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Let $V$ be a vector space and $U$ a fixed subspace of $V$. We denote the semigroup of all linear transformations on $V$ under composition of functions by $L(V)$. In this paper, we study the semigroup of all linear transformations on $V$ whose restrictions belong to the general linear group $GL(U)$, denoted by $L_{GL(U)}(V)$. More precisely, we consider the subsemigroup \[ L_{GL(U)}(V)=\{\alpha\in L(V):\alpha|_U\in GL(U)\} \] of $L(V)$. In this work, Green's relations and ideals of this semigroup are described. Then we also determine the minimal ideal and the set of all minimal idempotents of it. Moreover, we establish an isomorphism theorem when $V$ is a finite dimensional vector space over a finite field. Finally, we find its generating set.

Let $T(X)$ be the full transformation semigroup on a set $X$ under the composition of functions. For any equivalence relation $E$ on $X$, define a subsemigroup $T_{E^*}(X)$ of $T(X)$ by $$T_{E^*}(X)=\{\alpha\in T(X):\text{for all}\ x,y\in X, (x,y)\in E\Leftrightarrow (x\alpha,y\alpha)\in E\}.$$ In this paper, we show that the regular part of $T_{E^*}(X)$, denoted $\mathrm{Reg}(T)$, is the largest regular subsemigroup of $T_{E^*}(X)$. Then its Green's relations and ideals are described. Moreover, we find the kernel of $\mathrm{Reg}(T)$ which is a right group and can be written as a union of symmetric groups. Finally, we prove that every right group can be embedded in that kernel., Comment: 12 pages, part of this work was presented in the Workshop on General Algebra / Arbeitstagung Allgemeine Algebra (AAA102), University of Szeged, Hungary, June 24-26, 2022

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 Universalis

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 Universalis