Sandwich Semigroups Research Articles

Sandwich semigroups and Brandt semigroups

Sandwich semigroups and Brandt semigroups

Sandwich semigroups in diagram categories

This paper concerns a number of diagram categories, namely the partition, planar partition, Brauer, partial Brauer, Motzkin and Temperley–Lieb categories. If [Formula: see text] denotes any of these categories, and if [Formula: see text] is a fixed morphism, then an associative operation [Formula: see text] may be defined on [Formula: see text] by [Formula: see text]. The resulting semigroup [Formula: see text] is called a sandwich semigroup. We conduct a thorough investigation of these sandwich semigroups, with an emphasis on structural and combinatorial properties such as Green’s relations and preorders, regularity, stability, mid-identities, ideal structure, (products of) idempotents, and minimal generation. It turns out that the Brauer category has many remarkable properties not shared by any of the other diagram categories we study. Because of these unique properties, we may completely classify isomorphism classes of sandwich semigroups in the Brauer category, calculate the rank (smallest size of a generating set) of an arbitrary sandwich semigroup, enumerate Green’s classes and idempotents, and calculate ranks (and idempotent ranks, where appropriate) of the regular subsemigroup and its ideals, as well as the idempotent-generated subsemigroup. Several illustrative examples are considered throughout, partly to demonstrate the sometimes-subtle differences between the various diagram categories.

Sandwich semigroups in locally small categories II: transformations

Fix sets X and Y, and write $$\mathcal P\mathcal T_{XY}$$ for the set of all partial functions $$X\rightarrow Y$$ . Fix a partial function $${a:Y\rightarrow X}$$ , and define the operation $$\star _a$$ on $$\mathcal P\mathcal T_{XY}$$ by $$f\star _ag=fag$$ for $$f,g\in \mathcal P\mathcal T_{XY}$$ . The sandwich semigroup $$(\mathcal P\mathcal T_{XY},\star _a)$$ is denoted $$\mathcal P\mathcal T_{XY}^a$$ . We apply general results from Part I to thoroughly describe the structural and combinatorial properties of $$\mathcal P\mathcal T_{XY}^a$$ , as well as its regular and idempotent-generated subsemigroups, $${\text {Reg}}(\mathcal P\mathcal T_{XY}^a)$$ and $$\mathbb E(\mathcal P\mathcal T_{XY}^a)$$ . After describing regularity, stability and Green’s relations and preorders, we exhibit $${\text {Reg}}(\mathcal P\mathcal T_{XY}^a)$$ as a pullback product of certain regular subsemigroups of the (non-sandwich) partial transformation semigroups $$\mathcal P\mathcal T_X$$ and $$\mathcal P\mathcal T_Y$$ , and as a kind of “inflation” of $$\mathcal P\mathcal T_A$$ , where A is the image of the sandwich element a. We also calculate the rank (minimal size of a generating set) and, where appropriate, the idempotent rank (minimal size of an idempotent generating set) of $$\mathcal P\mathcal T_{XY}^a$$ , $${\text {Reg}}(\mathcal P\mathcal T_{XY}^a)$$ and $$\mathbb E(\mathcal P\mathcal T_{XY}^a)$$ . The same program is also carried out for sandwich semigroups of totally defined functions and for injective partial functions. Several corollaries are obtained for various (non-sandwich) semigroups of (partial) transformations with restricted image, domain and/or kernel.

Sandwich semigroups in locally small categories I: foundations

Fix (not necessarily distinct) objects i and j of a locally small category S, and write $$S_{ij}$$ for the set of all morphisms $$i\rightarrow j$$ . Fix a morphism $$a\in S_{ji}$$ , and define an operation $$\star _a$$ on $$S_{ij}$$ by $$x\star _ay=xay$$ for all $$x,y\in S_{ij}$$ . Then $$(S_{ij},\star _a)$$ is a semigroup, known as a sandwich semigroup, and denoted by $$S_{ij}^a$$ . This article develops a general theory of sandwich semigroups in locally small categories. We begin with structural issues such as regularity, Green’s relations and stability, focusing on the relationships between these properties on $$S_{ij}^a$$ and the whole category S. We then identify a natural condition on a, called sandwich regularity, under which the set $${\text {Reg}}(S_{ij}^a)$$ of all regular elements of $$S_{ij}^a$$ is a subsemigroup of $$S_{ij}^a$$ . Under this condition, we carefully analyse the structure of the semigroup $${\text {Reg}}(S_{ij}^a)$$ , relating it via pullback products to certain regular subsemigroups of $$S_{ii}$$ and $$S_{jj}$$ , and to a certain regular sandwich monoid defined on a subset of $$S_{ji}$$ ; among other things, this allows us to also describe the idempotent-generated subsemigroup $$\mathbb E(S_{ij}^a)$$ of $$S_{ij}^a$$ . We also study combinatorial invariants such as the rank (minimal size of a generating set) of the semigroups $$S_{ij}^a$$ , $${\text {Reg}}(S_{ij}^a)$$ and $$\mathbb E(S_{ij}^a)$$ ; we give lower bounds for these ranks, and in the case of $${\text {Reg}}(S_{ij}^a)$$ and $$\mathbb E(S_{ij}^a)$$ show that the bounds are sharp under a certain condition we call MI-domination. Applications to concrete categories of transformations and partial transformations are given in Part II.

Transformation Representations of Sandwich Semigroups

ABSTRACTLet a be an element of a semigroup S. The local subsemigroup of S with respect to a is the subsemigroup aSa of S. The variant of S with respect to a is the semigroup with underlying set S and operation ⋆a defined by x⋆ay = xay for x, y ∈ S. We show that the following classes contain precisely the same semigroups, up to isomorphism: all local subsemigroups of all finite full transformation semigroups; and all variants of all finite full transformation semigroups. This result was discovered as a result of some experiments (and accidents) when working with the Semigroups package for GAP.

Semigroups of rectangular matrices under a sandwich operation

Let $\mathcal M_{mn}=\mathcal M_{mn}(\mathbb F)$ denote the set of all $m\times n$ matrices over a field $\mathbb F$, and fix some $n\times m$ matrix $A\in\mathcal M_{nm}$. An associative operation $\star$ may be defined on $\mathcal M_{mn}$ by $X\star Y=XAY$ for all $X,Y\in\mathcal M_{mn}$, and the resulting \emph{sandwich semigroup} is denoted $\mathcal M_{mn}^A=\mathcal M_{mn}^A(\mathbb F)$. These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. In this article, we study $\mathcal M_{mn}^A$ as well as its subsemigroups $\operatorname{Reg}(\mathcal M_{mn}^A)$ and $\mathcal E_{mn}^A$ (consisting of all regular elements and products of idempotents, respectively), as well as the ideals of $\operatorname{Reg}(\mathcal M_{mn}^A)$. Among other results, we: characterise the regular elements, determine Green's relations and preorders, calculate the minimal number of matrices (or idempotent matrices, if applicable) required to generate each semigroup we consider, and classify the isomorphisms between finite sandwich semigroups $\mathcal M_{mn}^A(\mathbb F_1)$ and $\mathcal M_{kl}^B(\mathbb F_2)$. Along the way, we develop a general theory of sandwich semigroups in a suitably defined class of \emph{partial semigroups} related to Ehresmann-style "arrows only" categories, we hope this framework will be useful in studies of sandwich semigroups in other categories. We note that all our results have applications to the \emph{variants} $\mathcal M_n^A$ of the full linear monoid $\mathcal M_n$ (in the case $m=n$), and to certain semigroups of linear transformations of restricted range or kernel (in the case that $\operatorname{rank}(A)$ is equal to one of $m,n$).

The primitive matrices of sandwich semigroups of generalized circulant Boolean matrices

Let Gn(C) be the sandwich semigroup of generalized circulant Boolean matrices with the sandwich matrix C and GC (Jn) the set of all primitive matrices in Gn(C). In this paper, some necessary and sufficient conditions for A in the semigroup Gn(C) to be primitive are given. We also show that GC (Jn) is a subsemigroup of Gn(C).

On the maximal subgroup of the sandwich semigroup of generalized circulant Boolean matrices

Let n be a positive integer, and C n (r) the set of all n × n r-circulant matrices over the Boolean algebra B = {0, 1}, $$G_n = \bigcup\limits_{r = 0}^{n - 1} {C_n (r)} $$ . For any fixed r-circulant matrix C (C ≠ 0) in G n , we define an operation “*” in G n as follows: A * B = ACB for any A, B in G n , where ACB is the usual product of Boolean matrices. Then (G n , *) is a semigroup. We denote this semigroup by G n (C) and call it the sandwich semigroup of generalized circulant Boolean matrices with sandwich matrix C. Let F be an idempotent element in G n (C) and M(F) the maximal subgroup in G n (C) containing the idempotent element F. In this paper, the elements in M(F) are characterized and an algorithm to determine all the elements in M(F) is given.

The semigroup of incline Hall matrices

This paper investigates the semigroup of incline Hall matrices. An incline Hall matrix means an incline matrix which is greater than or equal to an incline invertible matrix. It is proved that any incline Hall matrix has index. Some necessary and sufficient conditions for an incline Hall matrix to be regular in the semigroup of incline Hall matrices are given and Green's relations on the such semigroup are described. Also the sandwich semigroup of incline Hall matrices is considered. The main results in the present paper generalize some previous results on Boolean matrices, fuzzy matrices and lattice matrices.

On sandwich semigroups in Boolean group algebras

Let BG be a Boolean group algebra, and let R be a nonzero element in BG, where G is an abelian group of order n. The sandwich semigroup of BG with a sandwich element R is denoted by BG( R). The purpose of this paper is to discuss Green's classes, idempotent elements, regular elements, and maximal subgroups for BG( R), and apply the obtained results to Boolean matrices.

Morita Equivalence for Factorisable Semigroups

Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context (S, R.,SPR,RQS,〈〉 , ⌈⌉) with 〈〉 and ⌈⌉ surjective. For a factorisable semigroup S, we denote ζS = {(s1, s2) ∈S×S|ss1 = ss2, ∀s∈S}, S' = S/ζS and US-FAct = {SM∈S− Act |SM = M and SHomS(S, M) ≅M}. We show that, for factorisable semigroups S and M, the categories US-FAct and UR-FAct are equivalent if and only if the semigroups S' and R' are strongly Morita equivalent. Some conditions for a factorisable semigroups to be strongly Morita equivalent to a sandwich semigroup, local units semigroup, monoid and group separately are also given. Moreover, we show that a semigroup S is completely simple if and only if S is strongly Morita equivalent to a group and for any index set I, S⊗SHomS(S, ∐i∈IS) →∐i∈IS, s⊗t·ƒ↦ (st)ƒ is an S-isomorphism.

The Semigroup of Hall Matrices over Distributive Lattices

of Hall matrices over a complete and completely distributive lattice L is studied. A Hall matrix is a matrix which is greater (for the order associated with the lattice structure) than an invertible matrix. Some necessary and sufficient conditions for a Hall matrix to be regular in the semigroup H n (L) are given and Green's relations of the semigroup H n (L) are described. Also, the sandwich semigroup of Hall matrices over the lattice L is studied.

On the Sandwich Semigroups of Group Boolean Matrices

Let GMn be the semigroup of all the n x n (n \geq 2)$ group Boolean matrices, and let R be a nonzero element in GMn, where G is an n-order Abelian group. The sandwich semigroup of GMn with the sandwich element R is denoted by GMn(R). The purpose of this paper is to discuss the Green's classes, idempotent elements, maximal subgroups, and regular elements in GMn(R). If G is an n-order cyclic group, our results are exactly the results of Wenchao Huang in [Linear Algebra Appl., 25 (1979), pp. 135--160].

Matrices which belong to an idempotent in a sandwich semigroup of circulant Boolean matrices

Let C n be the semigroup of all the n × n circulant Boolean matrices ( n ⩾ 2), and let R be a nonzero element in C n . The sandwich semigroup of C n ( R) is the set C n together with the multiplication rule A ∗ B = ARB. For a given idempotent I R in C n ( R), we characterize the matrices A ∈ C n ( R) that belong to I R in the sense that A∗ k = I R for some nonnegative integer k. When R is the identity matrix, the result specializes to a theorem of Schwarz.

Homomorphisms of sandwich semigroups and sandwich near-rings

Homomorphisms of sandwich semigroups and sandwich near-rings

Homomorphisms of sandwich semigroups and sandwich near-rings

Homomorphisms of sandwich semigroups and sandwich near-rings

On the sandwich semigroups of circulant Boolean matrices

Let C n be the semigroup of all the n × n ( n ⩾ 2) circulant Boolean matrices, and R a nonzero element in C n . The sandwich semigroup of C n with the sandwich element R is denoted by C n ( R). The purpose of this paper is to discuss the Green's classes, idempotent elements, regular elements, and maximal subgroups in C n ( R). In Section 4, a necessary and sufficient condition on the Green's class L R ( A) is given, where L R ( A) is the Green's class in the sandwich semigroup C n ( R) containing A, and A is an arbitrary circulant Boolean matrix in C n ( R). In Sections 5 and 6, the idempotent elements and the maximal subgroups containing an idempotent element in C n ( R) are discussed. Some necessary and sufficient conditions which characterize the idempotent elements and maximal subgroups are obtained. In Section 7, we use the results of Sections 5 and 6 to obtain a necessary and sufficient condition which characterizes the regular elements in C n ( R). Otherwise, some counting theorems about the idempotent elements, the regular elements, and the maximal subgroups are given.

Automorphism groups of sandwich semigroups

Automorphism groups of sandwich semigroups

Symons' $d$-congruence on sandwich semigroups

Symons' $d$-congruence on sandwich semigroups

Maximal groups in sandwich semigroups of binary relations

Maximal groups in sandwich semigroups of binary relations