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 $i\to j$. 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 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 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 $S_{ij}^a$, Reg$(S_{ij}^a)$ and $\mathbb E(S_{ij}^a)$; we give lower bounds for these ranks, and in the case of 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.Comment: 23 pages, 1 figure. V2: updated according to referee report, expanded abstract, to appear in Algebra Universali

Sandwich semigroups in locally small categories II: Transformations

Fix sets $X$ and $Y$, and write $\mathcal{PT}_{XY}$ for the set of all partial functions $X\to Y$. Fix a partial function $a:Y\to X$, and define the operation $\star_a$ on $\mathcal{PT}_{XY}$ by $f\star_ag=fag$ for $f,g\in\mathcal{PT}_{XY}$. The sandwich semigroup $(\mathcal{PT}_{XY},\star_a)$ is denoted $\mathcal{PT}_{XY}^a$. We apply general results from Part I to thoroughly describe the structural and combinatorial properties of $\mathcal{PT}_{XY}^a$, as well as its regular and idempotent-generated subsemigroups, Reg$(\mathcal{PT}_{XY}^a)$ and $\mathbb E(\mathcal{PT}_{XY}^a)$. After describing regularity, stability and Green's relations and preorders, we exhibit Reg$(\mathcal{PT}_{XY}^a)$ as a pullback product of certain regular subsemigroups of the (non-sandwich) partial transformation semigroups $\mathcal{PT}_X$ and $\mathcal{PT}_Y$, and as a kind of "inflation" of $\mathcal{PT}_A$, 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 $\mathcal{PT}_{XY}^a$, Reg$(\mathcal{PT}_{XY}^a)$ and $\mathbb E(\mathcal{PT}_{XY}^a)$. 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.Comment: 35 pages, 11 figures, 1 table. V2: updated according to referee report, expanded abstract, to appear in Algebra Universali

Partial oders on semigroups of partial transformations with restricted range

Let $X$ be any set and $P(X)$ the set of all partial transformations defined on $X$, that is, all functions $\alpha:A\to B$ where $A,B$ are subsets of $X$. Then $P(X)$ is a semigroup under composition. Let $Y$ be a subset of $X$. Recently, Fernandes and Sanwong defined $PT(X,Y)=\{\alpha\in P(X):X\alpha\subseteq Y\}$ and $I(X,Y)$ the set of all injective transformations in $PT(X,Y)$. So $PT(X,Y)$ and $I(X,Y)$ are subsemigroups of $P(X)$. In this paper, we study properties of the so-called natural partial order $\leq$ on $PT(X,Y)$ and $I(X,Y)$ in terms of domains, images and kernels, compare $\leq$ with the subset order, characterize the meet and join of these two orders, then find out elements of $PT(X,Y)$ and $I(X,Y)$ those are compatible. Also, the minimal and maximal elements are described. DOI: 10.1017/S000497271200002