Outer inverses in semigroups belonging to the prescribed Green’s equivalence classes

We provide existence criteria and characterizations for outer inverses in a semigroup belonging to the prescribed Green’s R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}$$\end{document}-, L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {L}}$$\end{document}- and H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {H}}$$\end{document}-classes. These results generalize the well-known problem of finding outer inverses of a matrix over a field with the prescribed range or/and null space. Outer inverses belonging to the prescribed Green’s R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}$$\end{document}- and L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {L}}$$\end{document}-classes also represent extensions of Drazin’s (b, c)-inverses and Mary’s inverses along an element. We provide an overview of other such extensions that have emerged recently and compare them with the extensions introduced in this paper.


Introduction
The study of generalized inverses has a very long and rich history.Penrose's approach to generalized inverses made it possible to extend the concept of the Moore-Penrose inverse (as well as certain related concepts) from matrices and operators to more general algebraic structures.As a result, various combinations of Moore-Penrose equations and related inverses were studied in the settings of (involutive) semigroups, rings, and other related structures.In addition to Moore-Penrose inverses, many other generalized inverses have been extensively studied, such as least-squares and minimum-norm inverses, the Drazin inverse, the group inverse, core and dual core inverses, and others.At the root of all these generalized inverses lie inner and outer inverses.As known, inner and outer inverses play an essential role in matrix theory.In particular, inner inverses play a key role in solving linear matrix equations and systems of linear equations.In addition, outer inverses are widely used in iterative methods for solving nonlinear equations, stable approximations of ill-posed problems, solving linear and nonlinear problems involving rank-deficient generalized inverses, statistics, and other areas.
Two new and important types of generalized inverses have emerged recently: the inverse along an element, introduced by Mary [44], and the (b, c)-inverse, introduced by Drazin [24].Both are outer inverses which, as their special cases, include the Moore-Penrose inverse, group inverse, Drazin inverse, core and dual core inverses, and other important types of generalized inverses.In addition, (b, c)-inverses originated as semigroup-theoretical counterparts of outer inverses of matrices with the prescribed range and null space (cf.[3, Ch. 2, Sect.6; Ch. 7, Sect.4]).
Mary [44] defined and studied inverses along an element using one of the most powerful tools of the semigroup theory-Green's relations.They have also been used in his other studies [45][46][47][48][49].However, to solve equations uniquely, Mary mainly emphasized relation H . Therefore, only a part of what the theory of Green's relations offers has been used in these papers.Our goal is to show the full strength of that theory here, notably by using the one-sided relations R and L .The general problem we discuss is the existence criteria and characterization of outer and inner inverses in semigroups belonging to the prescribed Green's R-, L -and H -classes.This problem generalizes the problem of determining the existence criteria and characterization of the outer inverse of matrices over a field with the prescribed range and/or null space.In addition, we discuss the existence criteria and characterization of inner inverses belonging to the prescribed principal right, left, and quasi-ideals.We recover that Drazin's (b, c)inverse and Mary's inverse along an element are essentially the same (see also [47, Proposition 1.4, Corollary 1.1], [48,75]).Both are outer inverses that belong to a prescribed Green's H -class, and the only difference is how that class is represented.
One of the ways we use here for setting criteria of existence of outer and inner inverses with prescribed properties is through solvability of certain equations in a semigroup, and the solutions of these equations are used to characterize these inverses.In most cases, the criteria are set using only one equation, and even when using a system of two equations, or in one case three, we show that they have the same sets of solutions, so it is enough to solve only one of them.This fact can be extremely important when generalized inverses of matrices are computed using neural networks.Such an approach emerged in the 1990s and has been gaining momentum in recent years.As pointed out in [74], the complexity of the neural network increases as the number of matrix equations increases, and decreasing the number of matrix equations needed to compute the desired generalized inverse will reduce this complexity.As a result, the architecture and implementation of the neural network will be simplified.
The paper is organized as follows.In Sect.2, we introduce basic concepts and notation concerning semigroups and generalized inverses and outline the main results that will be used in the further text, as well as examples that illustrate Green's relations in some important semigroups, including semigroups of matrices over a field and a semiring.Then, in Sect.3, we provide the existence conditions and characterizations for outer inverses of a given element belonging to the prescribed Green's Rand Lclasses, as well as for inner inverses belonging to the prescribed right and left principal ideals and reflexive generalized inverses belonging to the prescribed Green's Rand L -classes.In Sect.4, we show that Mary's inverse along an element, Drazin's (b, c)inverse, and Bott-Duffin (e, f )-inverse of a given element are just three different ways in representing the same thing-the outer inverse of this element belonging to the corresponding H -class, whereas the Djordjević and Wei's outer inverse with prescribed idempotents is their proper special case.We provide some new existence criteria and characterizations for (b, c)-inverses and inverses along an element, as well as for group inverses, and give new proofs of some known results concerning these inverses.In Sect.5, we give an overview of recently emerged extensions of (b, c)-inverses and inverses along an element and compare them with the concepts introduced in this paper.In Sect.6 we present computational consequences of the results obtained in the previous sections.In particular, we provide a table that links various types of equations with corresponding generalized inverses.Finally, in Sect.7 we give concluding remarks and visions of our future research.

Preliminaries: Green's equivalences, generalized inverses
This section introduces basic concepts and notation regarding semigroups and generalized inverses.For undefined notions and notation regarding semigroups we refer to [19,32], and for those regarding generalized inverses in different contexts we refer to [3,7,12,21,66].
It is worth noting that many of the statements proved here have their dual statements (see Table 2).For the sake of completeness, these dual statements will be stated, but their proofs will be omitted since any proof of a statement that has a dual can be transformed into a proof of its dual statement.
The results obtained in this paper are very general and can be applied to many specific mathematical systems such as partial and full mappings, linear operators, matrices over a field, ring or semiring, tensors, etc.
Let S is a semigroup.By S 1 we denote the semigroup S ∪ {1} arising from S by the adjunction of an identity element 1, unless S already has an identity, in which case S 1 = S.The smallest left ideal of S containing an element a ∈ S is S 1 a = Sa ∪ {a}, which is conveniently denoted by L(a) and called the principal left ideal of S generated by a. Analogously, the smallest right ideal of S containing a is aS 1 = aS ∪ {a}, and it is denoted by R(a) and called the principal right ideal of S generated by a, and the smallest ideal of S containing a is S 1 aS 1 = SaS ∪ aS ∪ Sa ∪ {a}, and it is denoted by J (a) and called the principal ideal of S generated by a. Recall that a quasi-ideal of S is a subset Q of S satisfying S Q ∩ QS ⊆ Q. Equivalently, Q is a quasi-ideal if and only if it is an intersection of a left and a right ideal of S. For any a ∈ S, the smallest quasi-ideal of S containing a, which is denoted by Q(a) and called the principal quasi-ideal of S generated by a, is represented by An equivalence L on S is defined by the rule: a L b if and only if a and b generate the same principal left ideal, i.e., if and only if L(a) = L(b).Similarly, equivalences R and J on S is defined by the rules: a R b if and only if a and b generate the same principal right ideal, i.e., if and only if R(a) = R(b), and a J b if and only if a and b generate the same principal ideal, i.e., if and only if J (a) = J (b).In other words, It is not hard to verify that L is a right congruence and R is a left congruence.The intersection of L and R is the equivalence denoted by H , i.e., H = L ∩ R. In addition, the relations L and R commute, that is, This analysis makes it possible to visualize the D-class as what Clifford and Preston [19] have called an 'eggbox', in which rows represent R-classes, columns represent L -classes, and cells represent H -classes (cf.Fig. 1a).
The restriction of a mapping φ to a subset X of its domain is denoted by φ| X .Let S be a semigroup and a ∈ S. The inner left translation λ a and the inner right translation  In addition, they preserve L -classes, i.e., x L λ u (x) and y L λ v (y), for all x ∈ R a , y ∈ R b .
Lemmas 2.1 and 2.2 are visualised by Fig. 1b and c, respectively.

Proposition 2.3
Every idempotent e of a semigroup S is a left identity for R e and a right identity for L e .In the case when ab ∈ R a ∩ L b , ab is called a trace product (cf.[44,51,56]).Before we give examples of Green's equivalences on some important types of semigroups, we introduce the notions of a dual semigroup, dual concept, and dual result.

Theorem 2.4 (Green's Theorem) If H is an H -class of a semigroup S, then either H
The dual semigroup of a semigroup S is a semigroup with the same carrier set S and the new multiplication * on S defined by a * b = b • a, for all a, b ∈ S. The dual of some semigroup-theoretical concept C is the concept obtained by replacing every product a • b in the definition of C by the product a * b = b • a.The dual statement d(A) of a statement A is the statement obtained by replacing each concept from A that has a dual with its dual concept.If a statement has the form "A implies B" then its dual has the form "d(A) implies d(B)".If one of them is true then so is the other, which is taken for granted without comment.
For better readability and comprehensibility of the paper, in Tables 1 and 2 we provide a comparative overview of the dual concepts used in this paper and the dual results that will be given later.

Dual concepts
Product

Left ideal Right ideal
Principal left ideal Principal right ideal Inner left translation λ a Inner right translation a In the sequel we present characterizations of Green's relations on some important semigroups.
Example 2.6 (Full transformation semigroup) Let T X be the full transformation semigroup on a set X , consisting of all maps from X into X .Subsemigroups of full transformation semigroups are called transformation semigroups.If X is a finite set with n elements, it will be represented as X = {1, 2, . . ., n}, and a map α ∈ T X will be represented as α = (α Although in the semigroup theory, it is usual for a mapping to be written on the right, in order to harmonize the notation with the one used in linear algebra, we will write mappings on the left.The same way of writing is also used in [30].The composition of two maps α, β ∈ T X , written multiplicatively, is then defined by (αβ)(x) = α(β(x)), for every x ∈ X .
According to [30,Theorem 4.5.1](see also [19, § 2.2] or [32, Exercise 2.6.16]) it follows By writing the mapping on the right, we also change the definition of multiplication in the full transformation semigroup and obtain the dual semigroup.In this case, Green's relations L and R interchange their roles, and for that reason, our characterizations of L and R in the full transformation semigroup differ from those given in [19,32].The same remark also applies to characterizations of L and R in the semigroup of linear transformations, considered in the following example.
Example 2.7 (Semigroup of linear transformations) Let L T (V ) be the semigroup of linear transformations of a vector space V over a field, under the multiplication defined in Example 2.6.For a linear transformation α ∈ L T (V ) we define the image or range im(α) as in Example 2.6, whereas the kernel or null space ker(α) is defined by ker(α) = {v ∈ V | α(v) = 0 }, and the rank of α is defined by rank (α) = dim(im(α)), i.e., as the dimension of the subspace im(α).
Based on [19,Exercise 2.2.6] or [32,Exercise 2.6.19],Green's equivalences on L T (V ) can be given by the same formulas as in (5).Note that the H -class of α is a group if and only if im(α) ∩ ker(α) = {0}.
It is essential to point out that in many sources, one can find the claim that matrices, except for the square ones, do not form a semigroup.However, this is only partially true.Matrices of arbitrary type form a partial semigroup.As we show in Example 2.8, utilizing the standard semigroup-theoretical method, it can be converted into a full semigroup without spoiling anything related to matrices.
Example 2.8 (Semigroup of matrices over a field) Let M(F) be the set of all matrices of an arbitrary type with entries in a field F, i.e., F m×n , let ∅ be an element which is not a member of M(F), and let M ∅ (F) = M(F) ∪ {∅}.Define the multiplication in M ∅ (F) such that the product in M ∅ (F) of two matrices coincides with their ordinary matrix product, if it is defined, and all other products are equal to ∅.With respect to such multiplication, M ∅ (F) is a semigroup with the zero ∅.We will call M ∅ (F) the semigroup of matrices with entries in F, and for the sake of convenience, we will call ∅ the empty matrix.Note that the above definition of the multiplication in M ∅ (F) is the standard procedure of the theory of semigroups for converting a partial semigroup into a full semigroup.Subsemigroups of the semigroup of matrices are called matrix semigroups.
The range R(A) and the null space N (A) of a matrix A ∈ F m×n ⊂ M ∅ (F) are defined by and rank(A) denotes the rank of A, i.e., the dimension of R(A) (cf.[3,66]).
It is easy to see that Example 2.9 (Semigroup of matrices over a semiring) Let S be a semiring and let M ∅ (S) be the semigroup of matrices with entries in S, defined in the same way as the semigroup M ∅ (F) from Example 2.8.
The row space R(A) of a matrix A ∈ S m×n is the span (set of all possible linear combinations) of its row vectors, and the column space C(A) of A is the span of its column vectors.Note that R(A) is considered as a subsemimodule of the (left) Ssemimodule S n (S 1×n ), and C(A) as a subsemimodule of the (right) S-semimodule S m (S m×1 ).The row rank of A, denoted by r (A), is the smallest possible cardinality of a spanning set for the row space, and the column rank of A, denoted by c (A), is the smallest possible cardinality of a spanning set for the column space (cf., e.g., [20,41]).Unlike matrices with entries in a field, the row rank and the column rank of a matrix with entries in a semiring are not necessarily equal.
For an arbitrary matrix A ∈ S m×n , any X ∈ S n× p induces an epimorphism of R(A) onto R(AX) sending u ∈ R(A) to uX , and any Y ∈ S q×m induces an epimorphism of As in Example 2.8, we have there exist matrices that induce mutually inverse isomorphisms between R(A) and R(B) ⇔ there exist matrices that induce mutually inverse isomorphisms between C(A) and C(B), A J B ⇔ there exist matrices that induce homomorphisms of R(A) into R(B), and vice versa ⇔ there exist matrices that induce homomorphisms of C(A) into C(B), and vice versa, (cf.[41]).Note that C(A) = C(B) implies A ∈ S m×n 1 and B ∈ S m×n 2 , for some m, n 1 , n 2 ∈ N, whereas R(A) = R(B) implies A ∈ S m 1 ×n and B ∈ S m 2 ×n , for some m 1 , m 2 , n ∈ N.
Next we proceed with generalized inverses.First we recall that an involutive semigroup is a semigroup S equipped with a unary operation * (called involution) satisfying (ab) * = b * a * and (a * ) * = a, for all a, b ∈ S. Depending on the context, in the sequel S will denote a semigroup or an involutive semigroup (in the cases when the Eqs.( 3) and (4) given below are taken into account).Let us consider the equations (1) axa = a, (2 where a ∈ S is a given element, and x is an unknown taking values in S. Let us note that Eqs. ( 3) and ( 4), as well as involutive semigroups, will not be considered in the rest of this paper.However, in the definitions we give here, we include these two equations to emphasize the significance of Eqs. ( 1) and ( 2) and their connection with fundamental types of generalized inverses-Moore-Penrose inverses, minimum-norm g-inverses and least-squares g-inverses (see the definitions below).This also applies to Eq. ( 5), which is very important for recently intensively studied core and dual core inverses.
For any γ ⊆ {1, 2, 3, 4, 5}, the system consisting of the equations (i), for i ∈ γ , is denoted by (γ ), and solutions to (γ ) are called γ -inverses of a.The set of all γ -inverses of a will be denoted by aγ , and the set of all γ -inverses of a contained in a set X will be denoted by aγ X .
Commonly, a {1}-inverse is called a g-inverse, a {2}-inverse is called an outer inverse, and a {1, 2}-inverse is called a reflexive inverse.A {1, 3}-inverse is known as a least-squares inverse, a {1, 4}-inverse is known as a minimum-norm inverse, a {1, 2, 3, 4}-inverse is known as the Moore-Penrose inverse or shortly MP-inverse of a, and a {1, 2, 5}-inverse is known as a group inverse of a.If a has at least one γ -inverse, then it is said to be γ -invertible.It is worth noting that an element of a semigroup having a {1}-inverse is called a regular element, and a semigroup whose every element is regular is called a regular semigroup.Examples of regular semigroups are the semigroup of matrices with entries in a field, the full transformation semigroup and the semigroup of linear transformations.In fact, the regularity of the full transformation semigroup of an infinite set and the semigroup of linear transformations of an infinitely dimensional vector space is a consequence of the Axiom of Choice.
For the sake of simplicity, the set of all {1}-invertible elements of a semigroup S will be denoted by S (1) (this should not be confused with the notation S 1 introduced at the beginning of Sect.2).In the theory of semigroups, this set is usually denoted by Reg(S), but here the first notation is more convenient.When we consider an arbitrary {1}-inverse of an element a ∈ S (1) , we usually denote it by a (1) .Similarly, the set of all {2}-invertible elements of S will be denoted by S (2) , and when we consider an arbitrary {2}-inverse of a ∈ S (2) , we usually denote it by a (2) .In the case of the existence, the Moore-Penrose inverse and the group inverse of an element a are unique, and they are denoted by a † and a # , respectively.The set of all idempotents contained in a subset X of a semigroup S will be denoted by X • .
It is worth noting that if T is a regular subsemigroup of a semigroup S, then any of Green's equivalences on T is simply the restriction of the corresponding Green's equivalence on S to T × T .In particular, if T is a regular subsemigroup of a full transformation semigroup, semigroup of linear transformations or a semigroup of matrices, then Green's equivalences L , R and H on T can be characterized in the same way as in Examples 2.6, 2.7 and 2.8, in terms of images/ranges and kernels/null spaces.
prescribed right and left principal ideals and reflexive generalized inverses belonging to the prescribed Green's R-classes and L -classes.
First, we state and prove the following theorem that can be viewed as the semigrouptheoretical counterpart of Theorem 1.3.7 [66] (called there the Urquhart formula; see also [63]).This theorem is fundamental as it provides, as we will see, both an existence criterion and a formula for calculating (b, c)-inverse.Theorem 3.1 Let S be a semigroup and let a, b, c ∈ S be such that cab ∈ S (1) , and let x = b(cab) (1) c, where (cab) (1) ∈ cab{1} is an arbitrary element.Then the following statements are true:

and only if ab and ca are trace products.
Proof (a) Let x ∈ a{1}, i.e., a = axa.Then and trivially, ab ∈ R(a) and ca ∈ L(a), so we obtain that ab R a and ca L a, that is, ab ∈ R a and ca ∈ L a .Conversely, let ab ∈ R a and ca ∈ L a .By ab R a it follows that a = abs, for some s ∈ S 1 .Since L is a right congruence and ca L a, it follows cab L ab.On the other hand, (cab) (1) cab L cab implies (cab) (1) cab L ab, i.e., (cab) (1) cab ∈ L ab .Now it can be concluded (cab) (1) cab is a right identity in L ab , whence ab(cab) (1) Therefore, axa = axabs = ab(cab) (1) cabs = abs = a, and we have proved that x ∈ a{1}.

Corollary 3.2
Let S be a semigroup and let a, b ∈ S be such that ab ∈ S (1) , and let x = b(ab) (1) .
where (ab) (1) ∈ ab{1} is an arbitrary element.Then the following statements are true: and only if ab is a trace product.

Corollary 3.3
Let S be a semigroup and let c, a ∈ S be such that ca ∈ S (1) , and let x = (ca) (1) c.where (ca) (1) ∈ ca{1} is an arbitrary element.Then the following statements are true: Next we provide the existence conditions and characterizations of outer inverses of a given element contained in the prescribed R-class.(v) ab ∈ S (1) and ab ∈ L b ; (vi) ab ∈ S (1) and b(ab) (1) ab = b, for some (equivalently every) (ab) (1) ∈ ab{1}.
Remark 3.6 Theorem 3.4 is a generalization of Theorem 3 [61], which concerns outer inverses of matrices with a prescribed range.There are two main differences between these two theorems.(a) The assumption ab ∈ S (1) appearing in Theorem 3.4 is not required in the case of matrices, since the semigroup of matrices is regular.This also holds when dealing with the full transformation semigroup and the semigroup of linear transformations.(b) One of the equivalent conditions of Theorem 3 [61] is N (AB) = N (B), whose semigroup-theoretical counterpart is the condition ab L b (that is, ab ∈ L b ) appearing in Theorem 3.4.In addition, Theorem 3 [61] contains the condition rank(AB) = rank(B), which is equivalent to N (AB) = N (B) (see [3,Chapter 1,Ex. 10] or [66,Theorem 1.1.3]).The condition concerning ranks is critical because in practice it is much easier to check the equality of ranks than the equality of null spaces.However, its semigroup-theoretical counterpart, the condition ab D b, is not present in Theorem 3.4, because in the general case it is not equivalent to ab L b.The equivalence of these two conditions is not valid, for example, in the full transformation semigroup of an infinite set and in the semigroup of linear transformations of an infinitely dimensional vector space, but it is true in the full transformation semigroup on a finite set.Namely, if X is a finite set, then for arbitrary α, β ∈ T X we have Example 3.7 Let us consider the full transformation semigroup T X on the set X = {1, 2, 3, 4}, and maps α, β ∈ T X given by α = (1242) and β = (1131).
Next, we provide the existence conditions and characterizations of outer inverses of a given element contained in a given L -class.These results generalize Theorem 5 [61].(iv) there exists v ∈ S such that c = cavc; (v) ca ∈ S (1) and ca ∈ R c ; (vi) ca ∈ S (1) and ca(ca) (1) c = c, for some (equivalently every) (ca) (1) ∈ ca{1}.
If these statements are true, then Example 3.9 Consider again the full transformation semigroup T X on the set X = {1, 2, 3, 4}, a map α ∈ T X given by α = (1242) (already considered in Example 3.7), and a map γ ∈ T X given by γ = (2424).
When working with inner inverses, it is interesting to locate them in prescribed principal right and left ideals, as seen in the next two theorems.These two theorems generalize Theorems 8 and 9 [61], and the same notes are valid as those given in Remark 3.6.
Let us recall that an element a of a semigroup S is called regular if there exists x ∈ S such that a = axa, i.e., if a has an inner inverse.Besides, a is called left regular if there exists x ∈ S such that a = xa 2 , and it is called right regular if there exists x ∈ S such that a = a 2 x.Applying the previously obtained results, we provide the following characterization of both regular and left regular elements.Theorem 3.10 Let S be a semigroup and a ∈ S. Then the following statements are equivalent: (i) there exists an outer inverse of a contained in the R-class R a ; (ii) there exists an inner inverse of a contained in the principal left ideal L(a); (iii) there exists a {1, 2}-inverse of a contained in the principal left ideal L(a); (iv) there exists x ∈ R(a) such that a = xa 2 ; (v) there exists u ∈ S such that a = aua 2 ; (vi) a is regular and left regular.
Further, the following representations hold under these equivalences: Proof The equivalence of statements (i)-(v) is obtained by applying Theorem 3.4 with b = a.The equivalence of (v) and (vi) is straightforward.Besides, ( 12) and ( 13) are obtained by putting b = a in ( 6) and (7).
A theorem dual to Theorem 3.10 is also valid.(iii) there exists u ∈ S such that a = a 2 ua; (iv) a is regular and right regular.
If these statements are true, then For more information on the relationship between the equations a = axa, a = xa 2 , a = axa 2 , a = a 2 x and a = a 2 xa, as well as their place in the implication diagram of all regularity types of elements of a semigroup defined by linear equations, we refer to [10].
Combining Theorems 3.4 and 3.8 we obtain the following theorem that provides the existence conditions and characterizations of a reflexive g-inverse of a given element contained in a given R-class.This theorem is a generalization of Theorem 10 [61].Theorem 3.12 Let S be a semigroup and a, b ∈ S. Then the following statements are equivalent: (i) there exists a {1, 2}-inverse of a contained in the R-class R b ; (ii) there exist u, v ∈ S such that b = buab and a = abva; (iii) there exists w ∈ S such that b = bwab and a = abwa; (iv) there exist s, t ∈ S such that a = abs and b = tab; (v) ab is a trace product; (vi) ab ∈ S (1) , ab(ab) (1) a = a and b(ab) (1) ab = b, for some (equivalently every) (ab) (1) ∈ ab{1}.
Suppose now that the equivalent statements (i)-(vi) hold.Then the equivalent statements of Theorems 3.4 and 3.8 hold, so b is regular, R(b) = bS and a{2} R b = {b(ab) (1) | (ab) (1) Since the opposite inclusion is clear, we conclude that ( 16) is fulfilled.
The equivalence (i)⇔(v) has been proven in a different form by Xu and Benítez [75,Theorem 3.14] in the context of rings.Remark 3. 13 According to the previous theorem, a necessary and sufficient condition for the existence of a {1, 2}-inverse of the element a contained in the R-class R b is the solvability of both equations b = buab and a = abva, where u and v are unknowns taking values in the semigroup S.However, when both equations are solvable, according to the proof of the implication (ii)⇒(iii) of the previous theorem, both equations have the same set of solutions.Therefore, if any of the equivalent conditions of Theorem 3.12 is satisfied, to compute a {1, 2}-inverse of a contained in R b it is enough to find a solution only to one of these two equations.
Due to duality, the following statement is also true.Theorem 3.14 Let S be a semigroup and a, c ∈ S. Then the following statements are equivalent: (i) there exists a {1, 2}-inverse of a contained in the L -class L c ; (ii) there exist u, v ∈ S such that c = cauc and a = avca; (iii) there exists w ∈ S such that c = cawc and a = awca; (iv) there exist s, t ∈ S such that a = cas and c = tca; (v) ca is a trace product; (vi) ca ∈ S (1) , ca(ca) (1) c = c and a(ca) (1) ca = a, for some (every) (ca) (1) ∈ ca{1}.
If these equivalences are true, then We end this section with the following example.
According to Theorem 2. In contrast, the equation ABV A = A is not solvable, since rank(AB) = 2 = rank(A) = 3, so R(AB) = R(A) does not hold.Therefore, there is a {2}-inverse of A with the range R(B), i.e., in the R-class of B, but there is no a {1, 2}-inverse nor a {1}-inverse of A with this range.

Outer and inner inverses belonging to the prescribed Green's H -classes
Let S be a semigroup and a, b, c ∈ S. According to Drazin [24], an element x ∈ S is called a (b, c)-inverse of a if it satisfies If a has a (b, c)-inverse, we say that it is (b, c)-invertible.It can be easily verified that, if the condition (D2) is fulfilled, the condition (D1) can be replaced by a simpler condition x ∈ bS ∩ Sc, or by a condition Drazin in [24] proved some basic properties of (b, c)-inverses, including the fact that the (b, c)-inverse of a is its outer inverse.Actually, Proposition 6.1 from [24] can be stated as follows.

Theorem 4.1 Let S be a semigroup and a, b, c ∈ S. An element x ∈ S is a (b, c)inverse of a if and only if it is an outer inverse of a contained in the H
The previous theorem can also be obtained from Theorems 3.4 and 3.8.A very important consequence of Theorem 4.1, i.e.Proposition 6.1 [24], is the following one:

Corollary 4.2 Let S be a semigroup and a, b, c, d, x ∈ S. Then (a) x is a (b, c)-inverse of a if an only if it is a (u, v)-inverse of a for some (equivalently all) u ∈ R b , v ∈ L c ; (b) x is a (b, c)-inverse of a if and only if it is a (d, d)-inverse of a for some (equivalently every) d ∈ R b ∩ L c . (c) x is a (d, d)-inverse of a if and only if it is an outer inverse of a contained in the H -class H d .
In view of Theorem 4.1 and Example 2.8, Drazin's (b, c)-inverse in a semigroup is an immediate generalization of the outer inverse of a matrix with a prescribed range and null space.Some related concepts have also been discussed in the literature on generalized inverses.In the sequel, we show that Drazin's (b, c)-inverse, Mary's inverse along an element and Bott and Duffin's (e, f )-inverse are essentially the same concepts, and that Djordjević and Wei's outer inverse with prescribed idempotents is their particular proper case.The equivalence of Mary's inverse along an element, Drazin's (b, c)-inverse and Bott and Duffin's (e, f )-inverse has been established recently in [48,Theorem C.4].   [24] (see also [11]), for a ∈ S and idempotents e, f ∈ S • , an element x ∈ S is a Bott-Duffin (e, f )-inverse of a if Drazin's (b, c)-inverse is essentially the same as Bott-Duffin (e, f )-inverse corresponding to a pair (e, f ) of idempotents of a semigroup S. As noted by Drazin in [24], Bott-Duffin (e, f )-inverse is Drazin's (b, c)-inverse for (b, c) = (e, f ).Moreover, if x is Drazin's (b, c)-inverse of a, then it is also Bott-Duffin (xa, ax)-inverse of a.In fact, Drazin's (b, c)-inverse is Bott-Duffin (e, f )-inverse for any e ∈ R b and f ∈ L c , and in particular, it is Bott-Duffin (bb (1) , c (1) c)-inverse for every b (1) ∈ b{1} and c (1) ∈ c{1}.Note that b (1) and c (1) exist because each element of the D-class containing b, c, cab and x is regular, whenever there is a (b, c)-inverse x of a.
The relationship between Bott-Duffin (e, f )-inverse and inverse along an element is described in [48,Theorem C.4]. Remark 4.5 (Djordjević and Wei's outer inverse with prescribed idempotents) For a ring R with identity 1, a ∈ R and p, q ∈ R • , Djordjević and Wei [23] defined a ( p, q)-outer inverse of a as an element x ∈ R satisfying We also call x an outer inverse of a with prescribed idempotents p and 1 − q.This concept can be easily transmitted into the context of semigroups, replacing the pair ( p, 1 − q) by a pair (e, f ) of idempotents of a semigroup S. If there is an outer inverse of a ∈ S with prescribed idempotents e and f , it is unique (cf.[23]).Unlike Mary's concept of an inverse along an element, Bott-Duffin's concept of an (e, f )-inverse, and Drazin's concept of a (b, c)-inverse, which are mutually equivalent, Djordjević and Wei's outer inverses with prescribed idempotents form their proper special case.Namely, if x is an outer inverse of a with prescribed idempotents e and f , then it is easy to check that x is Bott-Duffin (e, f )-inverse of a, and hence, it is Drazin's (e, f )-inverse of a.The converse does not hold.As Example 4.6 shows, if x is Drazin's (e, f )-inverse of a, for some idempotents e, f ∈ S • , then it is not necessary an outer inverse of a with prescribed idempotents e and f .Example 4.6 Consider the matrix A ∈ C 3×3 and its Moore-Penrose inverse X = A † , which are given by Then the projectors AX and X A are given by Further, consider idempotent matrices E and F represented by It is easy to verify that X AE = E and E X A = X A, so E R X A R X , and AX F = AX and F AX = F, whence F L AX L X .Thus, X R E and X L F, so X is the Drazin's (E, F)-inverse of A. In contrast, X differs from the outer inverse of A with prescribed idempotents E and F, since E = X A and F = AX.
It is known that Drazin's (b, c)-inverse and Mary's inverse along an element are unique, whenever they exist.The next theorem shows that their uniqueness can be derived from the well-known properties of classes of Green's equivalences stated in Green's theorem and lemmas.

Theorem 4.7 An element of a semigroup can have at most one outer inverse contained in a given H -class.
Proof Let S be a semigroup and a ∈ S. Suppose that x, y ∈ S are outer inverses of a such that x H y. Since L is a right congruence, xa L ya is implied by x L y. On the other hand, x R y, x R xa and y R ya yield xa R ya.Thus, xa H ya, and since both xa and ya are idempotents, we conclude xa = ya.Finally, according to Green's lemma, the mapping a | L x is a bijection of L x onto L xa , and by a (x) = a (y) it follows x = y.Now we state the existence criteria for (b, c)-inverses.Equivalence of statements (i), (ii) and (vi) has been proved by Drazin [24], while (ix), (x) and (xi) come from Mary [44] and Mary and Patrício [49].For the sake of completeness, we quote all these statements and give different proofs.(vii) cab ∈ S (1) , cab(cab) (1) c = c and b(cab) (1) cab = b, for some (equivalently every) (cab) (1) ∈ cab{1}; (viii) ab, ca ∈ S (1) , b(ab) (1) ab = b, ca(ca) (1) c = c and b(ab) (1) = (ca) (1) c, for some (ab) (1) ∈ ab{1} and (ca) (1) ∈ ca{1}; Proof The equivalence of statements (i) and (ii) was proved by Drazin in [24] (see also [25]), and (vi) is clearly just another way to write (ii).However, we will give an immediate, illustrative proof for (vi)⇒(i).Set x = ucabv.Then (iv)⇒(iii) and (iii)⇒(ii).These implications are clear.
(vii)⇒(ii).This implication can be simply verified.The equivalence (i)⇔(ix) is actually a result of Mary and Patricio (Theorem 2.2 [49]), and it can be easily derived as a consequence of the equivalence (i)⇔(vi) from this theorem and of Corollary 4.2.On the other hand, the equivalence of the statements Fig. 3 Visualisation of the proof of (vi)⇒(i) of Theorem 4.8 (i), (x) and (xi) has been proved by Mary (Theorem 7 [44]), but here we give a different proof, whose parts will be used in the further work.
The equivalence of statements (i) and (ix) of the previous theorem can also be stated as follows.

Corollary 4.9 [44] Let S be a semigroup and a, d ∈ S. Then there exists an outer inverse of a contained in the H -class H d if and only if dad H d.
Comments similar to those given in Remark 3.6 can also be given for Theorem 4.8.We will give some more comments concerning this theorem.Remark 4.10 (a) As we have seen, one of the necessary and sufficient conditions for the existence of a (b, c)-inverse of an element a is the solvability of both equations b = vcab and c = cabu, where u and v are unknowns taking values in the semigroup S.However, as noted by Drazin (Remark 2.3 in [24]), if the existence conditions are satisfied, to compute the (b, c)-inverse of a it is enough to solve only one of these two equations.Indeed, if v is an arbitrary solution of the first equation, then the (b, c)inverse of a is equal to vc, and of u is an arbitrary solution of the second one, then the (b, c)-inverse of a is equal to bu.
(b) Another necessary and sufficient condition for the existence of a (b, c)-inverse of a is the solvability of both equations b = bucab and c = cabvc, where u and v are unknowns taking values in S.However, if both equations are solvable, then they have the same set of solutions.Indeed, for random solution u to the first equation and random solution v to the second one we have buc = bucabvc = bvc, whence b = bucab = bvcab and c = cabvc = cabuc.In this case, for an arbitrary solution u to these two equations, the (b, c)-inverse of a is equal to buc.
(c) It is also worth noting that the conditions bu = vc in (v) and b(ab) (1) = (ca) (1) c in (viii) connect Theorems 3.4 and 3.8, in the sense that they provide the existence of an outer inverse of a which belongs both to R b and L c .These conditions are necessary because of Example 4.11 given below shows that there are cases in which there exist outer inverses of a in R b and L c , but not in R b ∩ L c .
The following set identities are valid: , ( 1133), (1313)}, Therefore, there exist outer inverses of α contained in R β and L γ , but there is no an outer inverse of α contained in R β ∩ L γ .
The next theorem provides our first representation of (b, c)-inverses.
Thus, b(cab) (1) c ∈ R b ∩ L c , and according to Theorem 4.1 we have that b(cab) (1) c is a (b, c)-inverse of a.
The representation (21) one obtains from Theorems 3.4 and 3.8.
The same representation of the (b, c)-inverse has been given in [75,Theorem 3.5].
The next theorem can be viewed as another formulation of Theorem 7 [44].However, here we give a different proof, based on the previously proved theorems.
Theorem 4.14 Let S be a semigroup and a, b, c ∈ S, and let x ∈ S be a (b, c)-inverse of a. Then for every d ∈ R b ∩ L c it follows ad ∈ H ax , da ∈ H xa , and Proof By the proof of (i)⇒(x) of Theorem 4.8, ad is group invertible, i.e., ad ∈ H e , for some e ∈ S • , and x is a (d, e)-inverse of a. Since (ad) # = (ead) # ∈ ead{1}, by Theorem 4.13 it follows In the same way we prove that x = (da) # d.

123
Theorem 4.8 has provided the criteria for an outer inverse belonging to the intersection of an R-class and an L -class.The next theorem does the same for an inner inverse and the intersection of a principal right ideal and a principal left ideal.Theorem 4. 15 Let S be a semigroup and a, b, c ∈ S. Then the following statements are equivalent: (i) there exists an inner inverse of a contained in the quasi-ideal R(b) ∩ L(c); (ii) there exist an inner inverse of a contained in R(b) and an inner inverse of a contained in L(c); (iii) there exist u, v ∈ S such that a = abua and a = avca; (iv) there exists w ∈ S such that a = abwca; (v) ab, ca ∈ S (1) , ab(ab) (1) a = a and a(ca) (1) ca = a, for some (equivalently all) (ab) (1) ∈ ab{1} and (ca) (1) ∈ ca{1}.
(iv)⇒(i).Let w ∈ S such that a = abwca.Set y = bwc.Then it is clear that a = aya and y ∈ R(b) ∩ L(c).
Let the statements (i) The rest of the proof follows immediately by Theorems 3.4 and 3.8 and the proof of (iv)⇒(i) of this theorem.
Combining Theorems 4.8 and 4.15 we obtain the following result on the existence of a {1, 2}-inverse belonging to the intersection of an R-class and an L -class.The equivalence (i)⇔(iv) can be seen as the (b, c)-inverse version of [44,Corollary 9].Further, suppose that the statements (i)-(iii) are true.Let x be the unique {1, 2}inverse of a contained in R b ∩ L c .The proof of the first equality in (23) is included in the proof of (ii)⇒(i).Consider an arbitrary element u ∈ S satisfying one of the conditions b = bucab, c = cabuc and a = abuca.If b = bucab, by Theorem 4.8 and Remark 4.10, buc is the unique outer inverse of a contained in R b ∩ L c , whence x = buc.In the same way we show that if c = cabuc is satisfied, then x = buc.Finally, if a = abuca, then cab = cabucab, which means that u ∈ cab{1}, and Theorem 4.13 initiates x = buc.
Let us give several remarks concerning the previous theorem.it is necessary to fulfill some additional requirements that would link two sets of {2}inverses, such as the requirement bu = vc in (v) of Theorem 4.8 or b(ab) (1) = (ca) (1) c in (viii) of the same theorem.In contrast, the previous theorem shows that the existence of {1, 2}-inverses of a in R b and L c guarantees the existence of a {1, 2}-inverse of a in R b ∩ L c , without any additional requirements.(b) According to Theorem 4.16, a necessary and sufficient condition for the existence of a {1, 2}-inverse of a contained in R b ∩ L c is solvability of the system of equations b = bucab, c = cabuc, a = abuca, where u is an unknown taking values in S.However, if this system is solvable, then all individual equations have the same set of solutions.Indeed, if b = bucab, then buc is a (b, c)-inverse of a, so c = cabuc, and by the fact that buc is also an inner inverse of a it follows that a = abuca.In the same way we show that c = cabuc implies b = bucab and a = abuca.Finally, if a = abuca, then u ∈ cab{1} and again we obtain that buc is a (b, c)-inverse of a, whence b = bucab and c = cabuc.
(c) The theorem also provides two ways for computing the {1, 2}-inverse of a contained in the H -class R b ∩ L c , whenever it exists.The first one is based on solving any of the equations b = bucab, c = cabuc and a = abuca.In this case, we actually compute the {2}-inverse x of a contained in R b ∩ L c , and if the conditions of existence of a {1, 2}-inverse of a contained in R b ∩ L c are satisfied, then x must also be a {1}-inverse of a.
The second way is to compute any {1, 2}-inverse y of a in R b and any {1, 2}-inverse z of a in L c , and then the {1, 2}-inverse x of a contained in R b ∩ L c is computed as x = yaz.
On the other hand, for an arbitrary A ∈ S, if Consequently, for an arbitrary C ∈ S, if Let us note that S is a regular subsemigroup of M ∅ (R), and according to the remark given at the very end of Sect.2, Green's relations R, L and H on S can be represented by means of ranges and null spaces, as in Example 2.8.
The last result of this section provides existence conditions and characterizations of group inverses through outer and inner inverses belonging to prescribed Green's equivalence classes and solutions of certain linear equations in a semigroup.Theorem 4. 19 Let S be a semigroup and a ∈ S. Then the following statements are equivalent: (i) there exists an outer inverse of a contained in the H -class H a ; (ii) there exist an outer inverse of a contained in R a and an outer inverse of a contained in L a ; (iii) there exists an inner inverse of a contained in the quasi-ideal R(a) ∩ L(a); (iv) there exist an inner inverse of a contained in R(a) and an inner inverse of a contained in L(a); (v) there exists a {1, 2}-inverse of a contained in the R-class R a ; (vi) there exists a {1, 2}-inverse of a contained in the L -class L a ; (vii) there exist u, v ∈ S such that a = aua 3 and a = a 3 va; (viii) there exist u, v ∈ S such that a = aua 2 and a = a 2 va; (ix) there exists w ∈ S such that a = awa 2 and a = a 2 wa; (x) there exist s, t ∈ S such that a = a 2 s and a = ta 2 ; (xi) a is group invertible.
If these statements are true, then for arbitrary u, v, s, t ∈ S such that a = aua 2 , a = ava 3 , a = a 2 s and a = ta 2 , and arbitrary y ∈ a{2} R a and z ∈ a{2} L a .
Proof The equivalence of (ii), (iv) and (viii) follows by Theorems 3.10 and 3.11, the equivalence of (iii) and (iv) is obtained from Further, (x) is equivalent to a 2 ∈ H a , and according to Green's Theorem (Theorem 2.4), this is equivalent to the claim that H a is a group.This proves the equivalence of the statements (x) and (xi).
The implication (i)⇒(ii) is clear, and if a is group invertible, then a # is an outer inverse of a contained in H a , which means that (xi)⇒(i).
Further, suppose that the claims (i)-(xi) are true.By Theorems 3.12 and 3.14 we obtain a{2} R a = a{1, 2} R a and a{2} L a = a{1, 2} L a .Now, for any u ∈ S satisfying a = aua 2 it follows au ∈ a{2} R a = a{1, 2} R a , whence a = a(au)a = a 2 ua.If we set x = auaua, then it is easy to check that a = axa, x = xax and ax = xa, which means that a # = x = auaua.The remaining two equalities a # = ava, for v ∈ S such that a = ava 3 , and a # = yaz, for y ∈ a{2} R a = a{1, 2} R a and z ∈ a{2} L a = a{1, 2} L a , 123 follow directly from Theorem 4.16, by putting b = c = a.Finally, let s, t ∈ S such that a = a 2 s = ta 2 .It is easy to check that tas = a # and ta = as, whence a # = tas = t 2 a = as 2 .
Remark 4. 20 According to the proof of the previous theorem, if both equations a = aua 2 and a = a 2 va are solvable, then they have the same set of solutions and to compute the group inverse of a it is enough to solve only one of them.This also holds for equations a = aua 3 and a = a 3 va.

Various extensions of a (b, c)-inverse and an inverse along an element
As we have pointed out earlier, various properties of a (b, c)-inverse and an inverse along an element have been considered in different contexts in a number of recent papers.In addition, various extensions of these concepts have recently emerged, and in this section we will compare them with the concepts we have introduced in this paper.Bapat et al. [2] dealt with outer inverses x ∈ a{2} which satisfy x S ⊆ bS and Sx ⊆ Sc, where S is a semigroup and a, b, c ∈ S. One can easily show that they are outer inverses of a belonging to the quasi-ideal R(b) ∩ L(c), so the result from [2] regarding such outer inverses can be restated as follows.
Theorem 5.1 Let S be a semigroup and a, b, c ∈ S. Then the following statements are equivalent: (i) there exists an outer inverse of a contained in the quasi-ideal R(b) ∩ L(c); (ii) cab ∈ S (2) .If these statements are true, then a{2} R(b)∩L(c) = {b(cab) (2) c | (cab) (2) ∈ cab{2}}. ( Relaxing the conditions of the above theorem, we provide an existence condition and a characterization of outer inverses that belong to the principal right ideal R(b) generated by a given element b ∈ S. Theorem 5.2 Let S be a semigroup and a, b ∈ S. Then the following statements are equivalent: (i) there exists an outer inverse of a contained in the principal right ideal R(b); (ii) ab ∈ S (2) .If these statements are true, then a{2} R(b) = {b(ab) (2) | (ab) (2) ∈ ab{2}}.(26) other things, Zhu et al. [81] proved that an element a of a semigroup S is left invertible along an element d ∈ S if and only if d ∈ L(dad), or equivalently, d L dad, and that a is right invertible along d if and only if d ∈ R(dad), or equivalently, d R dad.These two results are direct generalizations of Mary's result given here as Corollary 4.9.
In the case when S is an involutive semigroup, Zhu et al. [81] proved that an element a is left invertible along a * if and only if it is right invertible along a * , which is also equivalent to the claim that a is MP-invertible.Also, any of these claims is equivalent to the claim that a * is left invertible (or right invertible) along a.
Basic properties of one-sided (b, c)-inverses were studied by Drazin [25], in the context of semigroups, and by Wang and Mosić [68] and Ke et al. [40], in the context of rings, whereas one-sided (b, c)-inverses of matrices were studied by Benítez et al. [6].In particular, for a semigroup S and a, b, c ∈ S, Drazin [25] noted that a is left (b, c)-invertible if and only if there is v ∈ S such that b = vcab, which is clearly equivalent to b L cab (see also [6,40]).Analogously, a is right (b, c)-invertible if and only if there exists u ∈ S such that c = cabu, which is equivalent to c R cab.An element a may have multiple left or right (b, c)-inverses, of which not all must be outer inverses of a, but if it has both left and right (b, c)-inverses, then they must be unique and equal to each other, in which case they are a unique (b, c)-inverse of a [25].Strongly left (b, c)-invertible elements, defined as left (b, c)-invertible elements whose every left (b, c)-inverse is its outer inverse, have been studied by Wang and Mosić [68].Wang and Mosić [68] and Wang et al. [69] have also studied one-sided extensions of core inverses in the context of rings.
It is interesting to note that in the context of matrices left and right invertibility along an element are mutually equivalent, and therefore, they are equivalent to invertibility along an element, but this does not hold for left and right (b, c)-invertibility (cf.[6]).
When the conjunction of Mary's conditions (M1) and (M2) is splitted into two new conditions, this could also been done in another way, replacing the places of the Clearly (M ) is essentially the same condition as (D ), and (M r ) is the same as (D r ).If x satisfies (D ), according to Theorem 3.8 it is concluded that x is an outer inverse of a contained in the L -class L c , and if x satisfies (D r ), according to Theorem 3.4, x is an outer inverse of a contained in the R-class R b .This conclusion means that our one-sided extensions of the concepts of a (b, c)-inverse and an inverse along an element, discussed in Theorems 3.4 and 3.8, are different from those proposed by Zhu et al. [81] and Drazin [25], and have the advantage that always are outer inverses of the underlying element and are more precisely localized.
In a series of papers (b, c)-inverses in rings and their extensions have been studied through annihilators.For any a ∈ R, where R is a ring (usually with an identity), the right annihilator a • of a and the left annihilator • a of a are sets defined as follows: It is easy to verify that a • is a right ideal and • a is a left ideal of R. Of course, in the context of semigroups, this definition can only be applied to semigroups with zero.However, the methodology common to working with annihilators in rings cannot be applied because it is mainly based on the use of subtraction.A concept of annihilators suitable for use in arbitrary semigroups has been recently proposed by Drazin [27].For a semigroup S and a ∈ S, the right annihilator rann(a) of a and the left annihilator lann(a) of a are equivalence relations on S 1 defined as follows: Strictly speaking, these concepts are not generalizations of annihilators in rings, but in a sense they are.Namely, in a ring R with identity these concepts are interchangeable due to the fact that for all a, r , s ∈ R the following is true Green's relations were extended in [50,55] to the equivalence relations L • , R • and H • as follows: two elements of a semigroup S are related by L • (resp.R • ) in S if and only if they are related by Green's relation L (resp.R) in some over semigroup of S, and [50] (see also [28,29,42]), the following equivalences are valid for arbitrary a, b ∈ S: It should be noted that in [50,55] and later papers, these relations were respectively denoted by L * , R * and H * , but here we use notation with circles because of the shown connection with annihilators.The relationship between principal left and right ideals and right and left annihilators is shown in the next proposition.(B) The condition (iii) cab ∈ S (1) and cab ∈ L • b ∩ R • c ; is equivalent to each of the six conditions of Theorem 4.8, and implies (i) and (ii).(C) If b, c ∈ S (1) , then each of the conditions (i) and (ii) implies (iii) and all the conditions of Theorem 4.8.

Proof (A)
The equivalence of the conditions (i) and (ii) follows directly from Theorems 5.8 and 5.5.
(B) Let the condition (i) of Theorem 4.8 hold, i.e., let there is x ∈ S such that xax = x and x ∈ R b ∩ L c .Then x, b, c and cab belong to the same D-class (see Fig. 3), and since x ∈ S (1) it follows that b, c, cab ∈ S (1) .Now, according to (vi) of Theorem 4. (B) The condition (iii) cab ∈ S (1) and cab ∈ L b ∩ R • c ; is equivalent to each of the six conditions of Theorem 4.8, and implies (i) and (ii).(C) If c ∈ S (1) , then each of the conditions (i) and (ii) implies (iii) and all the conditions of Theorem 4.8.The presented theoretical results provide equivalent conditions for the existence and corresponding characterizations for outer inverses.The results are stated in a very general form, as outer inverses in semigroups belonging to the prescribed Green's R-, L -and H -classes.They are applicable in many mathematical structures due to very general theoretical results.Particularly, the obtained results generalize the wellknown and frequently investigated problem of finding outer matrix inverse with the prescribed image or/and kernel in various spaces.Primarily, the corresponding investigations in numerical linear algebra and on linear operators can be expected.Further, the corresponding theoretical research and computational procedures are expectable for matrices with entries over a field, ring or semiring, time-varying matrices, multidimensional arrays, etc.So far, in [59], the authors applied this global algorithmic framework in the tensor case.
In our further research we will introduce the concept of trace factorization, which can be viewed as a semigroup-theoretical generalization of the full-rank factorization of matrices.Some additional characterizations of the group and (b, c)-invertibility in semigroups will be given using the trace factorization.Some new existence criteria, characterizations and representations for {1, 3}-and {1, 4}-inverses, MP-inverses, and (dual)core inverses will also be established.
The underlying equations can be solved using various methods.The GNN approach was used in [61].Of course, other computational techniques can be applied in solving required equations, leading to a number of various computational algorithms.
Moreover, there is a number of representations of various classes of generalized inverses which have not been investigated so far.These cases are marked by "no" in the last column of Table 3.This means that all equations corresponding to "no" define a new approach in computation of new classes of generalized inverses, which can be used in further research.
We emphasize that there is another way to turn the set of all matrices over a field (of arbitrary types) into a full semigroup, while preserving the classical matrix product.This can be done using the so-called semi-tensor product, an associative product that is defined for matrices of arbitrary types and coincides with the classical matrix product whenever it is defined (cf.[17]).However, such a semigroup will be the subject of our research that will be conducted in the future.
where, as is usual, • denotes the composition of relations, so D = L • R = R • L is the smallest equivalence on S containing both L and R (it is the join of L and R in the lattice of all equivalences on S).Since L ⊆ J and R ⊆ J , it follows D ⊆ J .Equivalences L , R, J , H and D are known as Green's equivalences.The L -class (resp.R-class, H -class, D-class) containing the element a will be denoted by L a (resp.R a , H a , D a ).Each D-class of a semigroup S is the union of L -classes contained in it and the union of R-classes contained in it.In general, the intersection of an L -class and an R-class may be empty, but the situation is different if these two classes are contained in the same D-class.Namely, for any a, b ∈ S we have

Lemma 2 . 1 (
Green's lemma) Let a and b be R-related elements of a semigroup S, and let s, t ∈ S 1 be elements such that as = b and bt = a.Then the right translations s | L a and t | L b are mutually inverse bijections from L a onto L b and L b onto L a , respectively.In addition, they preserve R-classes, i.e., x R s (x) and y R t (y), for all x ∈ L a , y ∈ L b .Lemma 2.2 (Green's lemma) Let a and b be L -related elements of a semigroup S, and let u, v ∈ S 1 be elements such that ua = b and vb = a.Then the left translations λ u | R a and λ v | R b are mutually inverse bijections from R a onto R b and R b onto R a , respectively.

Theorem 2 . 5 (
Miller-Clifford's theorem) Let S be a semigroup and let a, b ∈ S. Then ab ∈ R a ∩ L b if and only if R b ∩ L a contains an idempotent.

Fig. 2
Fig. 2 Visualisation of the situation considered in Theorem 3.1(d) (d) Let x ∈ a{1, 2} and x ∈ R b ∩ L c .Then by (c) and (a) it follows that cab ∈ R c ∩ L b , ab ∈ R a and ca ∈ L a .By ab R a and ca L a we obtain cab R ca and cab L ab, whence ca R c and ab L b. Thus, ab ∈ R a ∩ L b and ca ∈ R c ∩ L a , i.e., ab and ca are trace products.Conversely, let ab ∈ R a ∩ L b and ca ∈ R c ∩ L a .Then by ab R a and ca L a it follows cab R ca R c and cab L ab L b, and hence, cab ∈ R c ∩ L b .Now, the statements (a) and (c) imply x ∈ a{1, 2}.The situation considered in Theorem 3.1(d) is shown in Fig. 2. The next two corollaries are immediate consequences of Theorem 3.1.Namely, applying Theorem 3.1 to the semigroup S 1 , for a, b ∈ S and c = 1 we obtain Corollary 3.2, and for a, c ∈ S and b = 1 we obtain Corollary 3.3.

Theorem 3 . 4
Let S be a semigroup and a, b ∈ S. Then the following statements are equivalent: (i) there exists an outer inverse of a contained in the R-class R b ; (ii) there exists an inner inverse of b contained in the principal left ideal L(a); (ii') there exists a {1, 2}-inverse of b contained in the principal left ideal L(a); (iii) there exists x ∈ R(b) such that b = xab; (iv) there exists u ∈ S such that b = buab; Let v be an inner inverse of b contained in L(a), i.e., let b = bvb and v = sa, for some s ∈ S 1 .If s ∈ S, then b = buab for u = s, and if s = 1, then b = bab and b = buab for u = ab.(iv)⇒(ii).If u ∈ S such that b = buab, then ua ∈ b{1} ∩ L(a).

Theorem 3 . 8
Let S be a semigroup and a, c ∈ S. Then the following statements are equivalent: (i) there exists an outer inverse of a contained in the L -class L c ; (ii) there exists an inner inverse of c contained in the principal right ideal R(a); (ii') there exists a {1, 2}-inverse of c contained in the principal right ideal R(a); (iii) there exists x ∈ L(c) such that c = cax;

Theorem 3 .
11 Let S be a semigroup and a ∈ S. Then the following statements are equivalent:(i) there exists an outer inverse of a contained in the L -class L a ; (i) there exists an inner inverse of a contained in the principal right ideal R(a); (ii) there exists a {1, 2}-inverse of a contained in the principal right ideal R(a); (ii) there exists x ∈ L(a) such that a = a 2 x;

Example 3 . 15 (
Example in a semigroup of square matrices) Consider matrices A, B ∈ C 3×3 given by

Remark 4 . 3 (
Mary's inverse along an element) For two elements a and d of a semigroup S, Mary in[44] defined an inverse of a along d as an outer inverse of a contained in the H -class H d .Mary also pointed out that an inverse of a along an element d can equivalently be defined as an element x ∈ S which satisfies the following two conditions:(M1) x ∈ R(d) ∩ L(d); (M2)xad = d and dax = d.It has been proved in [48, Theorem C.4] that Mary's inverse along an element d of a semigroup is essentially the same as Drazin's (b, c)-inverse.Namely, Mary's inverse of a along d is Drazin's (b, c)-inverse of a for every pair b, c ∈ S such that H d = R b ∩ L c , while Drazin's (b, c)-inverse of a is Mary's inverse of a along d, for every d ∈ R b ∩ L c .Thus, the only difference between Drazin's and Mary's inverses lies in representing H -classes.In the case of a (b, c)-inverse, a H -class is given as the intersection of the R-class represented by b and the L -class represented by c, and in the case of an inverse along an element d, the same H -class is represented by its element d.

Theorem 4 . 8
Let S be a semigroup and a, b, c ∈ S. Then the following statements are equivalent: (i) there exists an outer inverse of a contained in the H -class R b ∩ L c .i.e., a is (b, c)-invertible; (ii) there exist u, v ∈ S satisfying b = vcab and c = cabu; (iii) there exist u, v ∈ S satisfying b = bucab and c = cabvc; (iv) there exists w ∈ S satisfying b = bwcab and c = cabwc; (v) there exist u, v ∈ S satisfying b = buab, c = cavc and bu = vc; (vi) cab ∈ R c ∩ L b ; for some (equivalently every) d ∈ R b ∩ L c ; (x) ad is group invertible and ad ∈ L d , for some (equivalently every) d ∈ R b ∩L c ; (xi) da is group invertible and da ∈ R d , for some (equivalently every) d ∈ R b ∩L c .
(vi)⇒(i).Let cab ∈ R c ∩ L b .Then b = ucab and c = cabv, for some u, v ∈ S 1 , and by Green's lemmas it follows that λ u | R c is a bijective mapping of R c onto R b preserving L -classes, and v | L b is a bijective mapping of L b onto L c preserving R-classes.
3), and by Theorem 4.1 we obtain that x is a (b, c)-inverse of a. (ii)⇒(v).If there exist u, v ∈ S such that b = vcab and c = cabu, then vc = vcabu = bu, whence b = vcab = buab and c = cabu = cavc.(v)⇒(iv).If there are u, v ∈ S such that b = buab, c = cavc and bu = vc, then bu = buabu and vc = vcavc, and the following conclusion follows for w = uav: b d and the left compatibility of R it follows that ax R ad.This means that e = ax is an idempotent in R ad , so it is a left identity in R ad , and therefore, ad = ead.Since d ∈ R b , and ax L x implies e = ax ∈ L x = L c , by Corollary 4.2 we obtain that x is a (d, e)-inverse of a, and by (i)⇔(vi) we conclude that ad = ead ∈ R e ∩ L d = R e ∩ L c = R e ∩ L e = H e .Thus, ad is group invertible and ad ∈ L d .(x)⇒(i).Suppose that ad is group invertible, i.e., ad ∈ H e , for some e ∈ S • , and let ad ∈ L d , for some d ∈ R b ∩ L c .Then ead = ad ∈ R e ∩ L d , and by (i)⇔(vi) we obtain that there exists x ∈ S such that x = xax and x ∈ R d ∩ L e .However, R d = R b and L e = L ad = L d = L c , so we conclude that x ∈ R b ∩ L c .Analogously we prove (i)⇒(xi) and (xi)⇒(i).

Theorem 4 .
16 Let S be a semigroup and a, b, c ∈ S. Then the following statements are equivalent: (i) there exists a {1, 2}-inverse of a contained in the H -class R b ∩ L c ; (ii) there exist a {1, 2}-inverse of a contained in R b and a {1, 2}-inverse of a contained in L c ; (iii) there exists u ∈ S such that b = bucab, c = cabuc and a = abuca; (iv) ca and ab are trace products.If all these statements are valid, then the unique {1, 2}-inverse x of a contained in the H -class R b ∩ L c can be represented by x = yaz = buc, (23) for arbitrary y ∈ a{2} R b = a{1} R(b) and z ∈ a{2} L c = a{1} L(c) , and an arbitrary u ∈ S satisfying b = bucab or c = cabuc or a = abuca.Proof The implication (i)⇒(ii) is clear.(ii)⇒(i).Suppose that there exist y ∈ a{1, 2} R b and z ∈ a{1, 2} L c , and set x = yaz.Then xax = yazayaz = yayaz = yaz = x and axa = ayaza = aza = a, which means that x ∈ a{1, 2}.On the other hand, x = yaz ∈ R(y) = R(b) and b = yab = yazab = xab ∈ R(x), and hence, R(x) = R(b), i.e., x ∈ R b .In the same way we obtain that x ∈ L c .Therefore, we have proved that x ∈ a{1, 2} R b ∩L c .(i)⇒(iii).Let there exists x ∈ a{1, 2} R b ∩L c .By Theorem 4.8 and Remark 4.10 it is concluded that x = buc, for some u ∈ S such that b = bucab and c = cabuc, and since x is a {1}-inverse of a we conclude a = axa = abuca.(iii)⇒(i).Let there exists u ∈ S such that b = bucab, c = cabuc and a = abuca.An application of Theorem 4.8 and Remark 4.10 lead to the conclusion buc ∈ a{2} R b ∩L c , and by a = abuca it follows that buc is a {1}-inverse of a.

Remark 4 .
17 (a) As we have seen in Example 4.11, the existence of {2}-inverses of a in R b and L c does not guarantee the existence of a {2}-inverse of a in R b ∩ L c , but

Example 4 . 18 = a 1 a 2 0 0 a 1 , a 2 ∈
Consider the set of matrices S R, a 1 = 0 .It is easy to check that S is a subsemigroup of the semigroup M ∅ (R) from Example 2.8 in which R = D = S × S and L = H , and for an arbitrary C ∈ S the L -class (H -class) of S containing C is given by the H -class of C. Hence, any matrix from S has a {1, 2}-inverse in each H -class of S.
Theorem 4.15 by putting b = c = a, and the equivalence of the statements (v)-(ix) is obtained from Theorems 3.12 and 3.14 by putting b = a and c = a.The implication (i)⇒(vii) is obtained from Theorem 4.8 by putting b = c = a, and the implication (vii)⇒(viii) is obvious.
equations xad = d and dax = d.This gives the conditions: (M ) x ∈ L(d) and dax = d; (M r ) x ∈ R(d) and xad = d.Similarly, the conjunction of Drazin's conditions (D1 ) and (D2) could been splitted such that the following conditions are obtained: (D ) x ∈ L(c) and cax = c; (D r ) x ∈ R(b) and xab = b.

Proposition 5 . 4 30 )Theorem 5 . 9
Let S be a semigroup and a, b ∈ S. Then: R(a) ⊆ R(b) ⇒ lann(b) ⊆ lann(a), L(a) ⊆ L(b) ⇒ rann(b) ⊆ rann(a).(If b is regular, then the reverse implications are also valid, i.e., R(a) ⊆ R(b) ⇔ lann(b) ⊆ lann(a), L(a) ⊆ L(b) ⇔ rann(b) ⊆ rann(a).(31) that the hybrid (b, c)-inverse or annihilator c)-inverse are genuine (b , c )-inverses, for some (any) regular elements b ∈ R • b and c ∈ L • c .All the mentioned results have been given in the context of rings, and in the next three theorems we state the corresponding results in the context of semigroups.Although these theorems are proved in an analogous way as the corresponding results in the context of rings, for the sake of completeness we provide proof of the first of these three theorems.Let S be a semigroup and a, b, c ∈ S. (A) The following two conditions are equivalent: (i) there exists an outer inverse of a contained in the H • -class R • b ∩ L • c ; (ii) there exists x ∈ S such that lann(b) ⊆ lann(x), rann(c) ⊆ rann(x), b = xab and c = cax.
8 and Proposition 5.4, it follows that cab ∈ L • b ∩ R • c .Conversely, let (iii) hold.Then cab = cabwcab, for some w ∈ S, whence (1, wcab) ∈ rann(cab) = rann(b) and (cabw, 1) ∈ lann(cab) = rann(c), which yields b = bwcab and c = cabwc.This means that the condition (iv) of Theorem 4.8 holds.Further, since (iii) (or equivalently, (i) of Theorem 4.8) implies the regularity of the elements x, b and c, according to Proposition 5.4 it follows that R b = R • b and L c = L • c .This means that (iii) implies (i).(C) The proof of this claim follows directly from Proposition 5.4 and the regularity of b and c.Theorem 5.10 Let S be a semigroup and a, b, c ∈ S. (A) The following two conditions are equivalent: (i) there exists an outer inverse of a contained in the R ∩ L • -class R b ∩ L • c ; (ii) there exists x ∈ S such that x ∈ R(b), rann(c) ⊆ rann(x), b = xab and c = cax.

Table 1
A comparative overview of the dual concepts used in this paper