Natural Partial Orders on Transformation Semigroups with Fixed Sets

Let X be a nonempty set. For a fixed subset Y of X, let FixX,Y be the set of all self-maps on X which fix all elements in Y. Then FixX,Y is a regular monoid under the composition of maps. In this paper, we characterize the natural partial order on Fix(X,Y) and this result extends the result due to Kowol and Mitsch. Further, we find elements which are compatible and describe minimal and maximal elements

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

Identities in the Algebra of Partial Maps

We consider the identities of a variety of semigroup-related algebras modelling the algebra of partial maps. We show that the identities are intimately related to a weak semigroup deductive system and we show that the equational theory is decidable. We do this by giving a term rewriting system for the variety. We then show that this variety has many subvarieties whose equational theory interprets the full uniform word problem for semigroups and consequently are undecidable. As a corollary it is shown that the equational theory of Clifford semigroups whose natural order is a semilattice is undecidable

Probabilistic Monads, Domains and Classical Information

Shannon's classical information theory uses probability theory to analyze channels as mechanisms for information flow. In this paper, we generalize results of Martin, Allwein and Moskowitz for binary channels to show how some more modern tools - probabilistic monads and domain theory in particular - can be used to model classical channels. As initiated Martin, et al., the point of departure is to consider the family of channels with fixed inputs and outputs, rather than trying to analyze channels one at a time. The results show that domain theory has a role to play in the capacity of channels; in particular, the (n x n)-stochastic matrices, which are the classical channels having the same sized input as output, admit a quotient compact ordered space which is a domain, and the capacity map factors through this quotient via a Scott-continuous map that measures the quotient domain. We also comment on how some of our results relate to recent discoveries about quantum channels and free affine monoids.Comment: In Proceedings DCM 2011, arXiv:1207.682