Defining relations for idempotent generators in finite full transformation semigroups

In 1966, John Howie showed that the semigroup $\mathcal{T}_{n}\setminus \mathcal{S}_{n}$ of all singular transformations on a n element set is generated by the set of all idempotent transformations of rank n?1. We give a presentation for $\mathcal{T}_{n}\setminus \mathcal{S}_{n}$ in terms of this generating set.     

Endomorphisms of finite full transformation semigroups

We describe all endomorphisms of finite full transformation semigroups and count their number.

     

Defining relations for idempotent generators in finite partial transformation semigroups

We give a presentation for the semigroup of all singular partial transformations on a finite set, in terms of the generating set consisting of all idempotent partial transformations of corank 1.     

Some remarks on finite full transformation semigroups

Communicated by J. M. Howie     

Products of idempotents in finite full transformation semigroups

Communicated by J. M. Howie     

Sequences with idempotent products from finite regular semigroups

We find the smallest integer (n) such that for every regular semigroup S of order n, every sequence of length (n) of elements of S contains a consecutive subsequence whose product is an -element, where = ‘idempotent’, ‘core’ and ‘subgroup and core’. For arbitrary semigroups of order n, we also find (n) where = ‘regular’, ‘group’, ‘core’, ‘regular and core’ and ‘subgroup and core’.     

Commutative cancellative semigroups of finite rank

The rank of a commutative cancellative semigroup S is the cardinality of a maximal independent subset of S. Commutative cancellative semigroups of finite rank are subarchimedean and thus admit a Tamura-like representation. We characterize these semigroups in several ways and provide structure theorems in terms of a construction akin to the one devised by T. Tamura for N-semigroups.     

Identities in full transformation semigroups

Presented by B. M. Schein.     

Graphs and finite transformation semigroups

Given a finite set X and a semigroup S of transformations of X, we study the orbitoids of S on X and on X² and, assuming S transitive, those of the statbilizer in S of an element α ∈ X. The action of S as a semigroup of endomorphisms of some relevant graphs (having X as vertex set) is also considered.     

Right ideals generated by an idempotent of finite rank

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let f(X₁,…,X_t) be an arbitrary and fixed polynomial over K in noncommuting indeterminates X₁,…,X_t with constant term 0 such that for some μK occurring in the coefficients of f(X₁,…,X_t). It is proved that a right ideal ρ of R is generated by an idempotent of finite rank if and only if the rank of f(x₁,…,x_t) is bounded above by a same natural number for all x₁,…,x_tρ. In this case, the rank of the idempotent that generates ρ is also explicitly given. The results are then applied to considering the triangularization of ρ and the irreducibility of f(ρ), where f(ρ) denotes the additive subgroup of R generated by the elements f(x₁,…,x_t) for x₁,…,x_tρ.     

On semigroups of endomorphisms of a chain with restricted range

Let X be a finite or infinite chain and let \({\mathcal{O}}(X)\) be the monoid of all endomorphisms of X. In this paper, we describe the largest regular subsemigroup of \({\mathcal{O}}(X)\) and Green’s relations on \({\mathcal{O}}(X)\) . In fact, more generally, if Y is a nonempty subset of X and \({\mathcal{O}}(X,Y)\) is the subsemigroup of \({\mathcal{O}}(X)\) of all elements with range contained in Y, we characterize the largest regular subsemigroup of \({\mathcal{O}}(X,Y)\) and Green’s relations on \({\mathcal{O}}(X,Y)\) . Moreover, for finite chains, we determine when two semigroups of the type \({\mathcal {O}}(X,Y)\) are isomorphic and calculate their ranks.     

Semigroup closures of finite rank symmetric inverse semigroups

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations I _λ ⁿ of the rank ≤ n is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.     

Rank functions of closure spaces of finite rank

We characterize those functions that are the rank functions of closure spaces of finite rank. In case such a function is defined on a finite set, we are able to improve this characterization.     

有限链上矩阵半群中幂等元的刻划

利用幂等元的定义以及矩阵的乘法研究了有限链K上的n×n阶矩阵的乘法半群Mn(K)中的幂等元,并给出若干刻划,主要是Simon结果的推广.从而,就可将Fuzzy矩阵中某些问题的研究转换为有限链K上矩阵中相应问题的研究,使其简单