Abstract:
We calculate the rank and idempotent rank of the semigroup $E(X,P)$ generated by the idempotents of the semigroup $T(X,P)$, which consists of all transformations of the finite set $X$ preserving a non-uniform partition $P$. We also classify and enumerate the idempotent generating sets of this minimal possible size. This extends results of the first two authors in the uniform case.

Abstract:
The rank of a semigroup is the cardinality of a smallest generating set. In this paper we compute the rank of the endomorphism monoid of a non-trivial uniform partition of a finite set, that is, the semigroup of those transformations of a finite set that leave a non-trivial uniform partition invariant. That involves proving that the rank of a wreath product of two symmetric groups is two and then use the fact that the endomorphism monoid of a partition is isomorphic to a wreath product of two full transformation semigroups. The calculation of the rank of these semigroups solves an open question.

Abstract:
Necessary and sufficient conditions are given for the endomorphism monoid of a profinite semigroup to be profinite. A similar result is established for the automorphism group.

Abstract:
Some of the classical results of Ramsey Theory can be naturally stated in terms of image partition regularity of matrices. There have been many significant results of image partition regular matrices as well as image partition regular matrices near zero. Here, we are investigating image partition regularity near an idempotent of an arbitrary Hausdorff semitopological semigroup (T, +) and a dense subsemigroup S of T . We describe some combinatorial applications on finite as well as infinite image partition regular matrices based on the Central Sets Theorem near an idempotent of T .

Abstract:
The study of the free idempotent generated semigroup $\mathrm{IG}(E)$ over a biordered set $E$ began with the seminal work of Nambooripad in the 1970s and has seen a recent revival with a number of new approaches, both geometric and combinatorial. Here we study $\mathrm{IG}(E)$ in the case $E$ is the biordered set of a wreath product $G\wr \mathcal{T}_n$, where $G$ is a group and $\mathcal{T}_n$ is the full transformation monoid on $n$ elements. This wreath product is isomorphic to the endomorphism monoid of the free $G$-act $F_n(G)$ on $n$ generators, and this provides us with a convenient approach. We say that the rank of an element of $F_n(G)$ is the minimal number of (free) generators in its image. Let $\varepsilon=\varepsilon^2\in F_n(G).$ For rather straightforward reasons it is known that if $\mathrm{rank}\,\varepsilon =n-1$ (respectively, $n$), then the maximal subgroup of $\mathrm{IG}(E)$ containing $\varepsilon$ is free (respectively, trivial). We show that if $\mathrm{rank}\,\varepsilon =r$ where $1\leq r\leq n-2$, then the maximal subgroup of $\mathrm{IG}(E)$ containing $\varepsilon$ is isomorphic to that in $F_n(G)$ and hence to $G\wr \mathcal{S}_r$, where $\mathcal{S}_r$ is the symmetric group on $r$ elements. We have previously shown this result in the case $ r=1$; however, for higher rank, a more sophisticated approach is needed. Our current proof subsumes the case $r=1$ and thus provides another approach to showing that any group occurs as the maximal subgroup of some $\mathrm{IG}(E)$. On the other hand, varying $r$ again and taking $G$ to be trivial, we obtain an alternative proof of the recent result of Gray and Ru\v{s}kuc for the biordered set of idempotents of $\mathcal{T}_n.$

Abstract:
Idempotents yield much insight in the structure of finite semigroups and semirings. In this article, we obtain some results on (multiplicatively) idempotents of the endomorphism semiring of a finite chain. We prove that the set of all idempotents with certain fixed points is a semiring and find its order. We further show that this semiring is an ideal in a well-known semiring. The construction of an equivalence relation such that any equivalence class contains just one idempotent is proposed. In our main result we prove that such an equivalence class is a semiring and find its order. We prove that the set of all idempotents with certain jump points is a semiring. 1. Introduction The idempotents play an essential role in the theory of finite semigroups and semirings. It is well known that in a finite semigroup some power of each element is an idempotent, so the idempotents can be taken to be like a generating system of the semigroup or the semiring. For deep results, using idempotents in the representation theory of finite semigroups, we refer the reader to [1, 2]. Let us briefly survey the contents of our paper. After the preliminaries, in Section 3 we show some facts about the fixed points of idempotent endomorphisms. The central result here is Theorem 9 where we prove that the set of all idempotents with fixed points , , is a semiring of order . Moreover, this semiring is an ideal of the semiring of all endomorphisms having at least as fixed points. In the next section we consider an equivalence relation on some finite semigroup such that for any follows if and only if , where and is an idempotent of . Then we consider the equivalence classes of semigroup which is, see [3], one subsemigroup of . Here we investigate the so-called jump points of the endomorphism and prove that between any two fixed points and of an endomorphism, where , there is a unique jump point. The main result of the paper is Theorem 19 where we prove that such an equivalence class is a semiring of order where is the th Catalan number. In the last section of the paper we consider idempotent endomorphisms with arbitrary fixed points but with certain jump points. Here we prove that the set of idempotent endomorphisms with identical jump points is a semiring. 2. Preliminaries We consider some basic definitions and facts concerning finite semigroups and that can be found in any of [1, 2, 4, 5]. As the terminology for semirings is not completely standardized, we say what our conventions are. An algebra with two binary operations + and？？ ？？on is called a semiring if(i) is a commutative

Abstract:
Idempotents yield much insight in the structure of finite semigroups and semirings. In this article, we obtain some results on (multiplicatively) idempotents of the endomorphism semiring of a finite chain. We prove that the set of all idempotents with certain fixed points is a semiring and find its order. We further show that this semiring is an ideal in a well known semiring. The construction of an equivalence relation such that any equivalence class contain just one idempotent is proposed. In our main result we prove that such equivalence class is a semiring and find his order. We prove that the set of all idempotents with certain jump points is a semiring.

Abstract:
An algebra $\A$ is said to be an independence algebra if it is a matroid algebra and every map $\al:X\to A$, defined on a basis $X$ of $\A$, can be extended to an endomorphism of $\A$. These algebras are particularly well behaved generalizations of vector spaces, and hence they naturally appear in several branches of mathematics such as model theory, group theory, and semigroup theory. It is well known that matroid algebras have a well defined notion of dimension. Let $\A$ be any independence algebra of finite dimension $n$, with at least two elements. Denote by $\End(\A)$ the monoid of endomorphisms of $\A$. We prove that a largest subsemilattice of $\End(\A)$ has either $2^{n-1}$ elements (if the clone of $\A$ does not contain any constant operations) or $2^n$ elements (if the clone of $\A$ contains constant operations). As corollaries, we obtain formulas for the size of the largest subsemilattices of: some variants of the monoid of linear operators of a finite-dimensional vector space, the monoid of full transformations on a finite set $X$, the monoid of partial transformations on $X$, the monoid of endomorphisms of a free $G$-set with a finite set of free generators, among others. The paper ends with a relatively large number of problems that might attract attention of experts in linear algebra, ring theory, extremal combinatorics, group theory, semigroup theory, universal algebraic geometry, and universal algebra.

Abstract:
The objective of this paper is to study the monoid of all partial transformations of a finite set that preserve a uniform partition. In addition to proving that this monoid is a quotient of a wreath product with respect to a congruence relation, we show that it is generated by 5 generators, we compute its order and determine a presentation on a minimal generating set.