Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Sandwich semigroups in locally small categories II: Transformations (1710.01891v3)

Published 5 Oct 2017 in math.GR, math.CT, and math.RA

Abstract: 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.

Summary

We haven't generated a summary for this paper yet.