Kind Of Semigroups Research Articles

Biunit pairs in semiheaps and associated semigroups

The notion of neutral element generalizes to a pair of elements in ternary algebras. Biunit pairs are introduced as pairs of elements in a semiheap that generalize the notion of Mal’cev element. In order to generalize the known correspondences between semiheaps and certain kinds of semigroups, families of functions generalizing involutions and conjugations, called switches and warps, are investigated. The main theorem establishes that there is a one-to-one correspondence between monoids equipped with a particular switch and semiheaps with a fixed biunit pair. This generalizes the celebrated result in semiheap theory that gives a one-to-one correspondence between involuted monoids and semiheaps with a fixed biunit element. A novel, previously undocumented, algebra is motivated by this result: diheaps are introduced as semiheaps whose elements belong to biunit pairs, which generalize the well-known case of heaps. Diheaps are of great interest since they are shown to be isomorphic to heaps only when they are heaps themselves and explicit non-heap examples are constructed from abelian groups and hypermatrices.

Good congruences on abundant semigroups with quasi-ideal adequate transversals

Abstract The aim of this paper is to study the congruences on abundant semigroups with quasi-ideal adequate transversals. The good congruences on an abundant semigroup with a quasi-ideal adequate transversal S° are described by the equivalence triple abstractly which consists of equivalences on the structure component parts I, S° and Λ. Also, it is shown that the set of all good congruences on this kind of semigroup forms a complete lattice.

Malcev product of superabundant semigroups and its subsemigroups

We use Malcev product of semigroups satisfying some axiomatic conditions to describe the structure of superabundant semigroups and some of its subclasses. Some characterization theorems of these kinds of semigroups are given.

Analysis and asymptotic stability of uniformly Lipschitzian nonlinear semigroup systems

In this paper, we shall study the asymptotic stability analysis of a special kind of semigroup on D × D, namely, the uniformly Lipschitzian semigroups.

Congruences on Split Quasi-Adequate Semigroups

In this paper, we give congruences on a split IC quasi-adequate semigroup by the congruence triple abstractly which consists of congruences on the structure component parts L, T, and R. We prove that the set of all congruences on this kind of semigroups is a complete lattice.

∗-congruences on abundant semigroups with SQ-adequate transversals

We give ∗-congruences on an abundant semigroup with an SQ-adequate transversal S ∘ by the ∗-congruence triple abstractly which consists of congruences on the structure component parts L, T and R. We prove that the set of all ∗-congruences on this kind of semigroups is a complete lattice.

Strong convergence theorems for a finite family of total asymptotically strict pseudocontractive semigroups in Banach spaces

The purpose of this paper is to introduce the concepts of total asymptotically strictly pseudocontractive semigroup, asymptotically strictly pseudocontractive semigroup etc., and to prove some strong convergence theorems of the explicit iteration process for these kinds of semigroups in arbitrary Banach spaces. The results presented in the paper extend and improve some recent results announced in the current literature.

Multi-valued totally quasi-ϕ-asymptotically nonexpansive semi-groups and strong convergence theorems in banach spaces

The purpose of this article is first to introduce the concept of multi-valued totally Quasiϕ-asymptotically nonexpansive semi-groups, which contains many kinds of semi-groups as its special cases, and then to modify the Halpern-Mann-type iteration algorithm for multi-valued totally Quasiϕ-asymptotically nonexpansive semi-groups to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in this article improve and extend the corresponding results announced by many authors recently.

The structure of medial weakly U-abundant semigroups

We consider the class of weakly U-abundant semigroups satisfying the congruence condition (C) containing both the class of regular semigroups and the class of abundant semigroups as its subclasses. The class of weakly U-abundant semigroups with a medial projection satisfying the congruence condition (C) will be particularly studied. This kind of semigroups will be called medial weakly U-abundant semigroups. In this paper, we establish a structure theorem for such semigroups. It is proved that every medial weakly U-abundant semigroup can be expressed by some kind of bands and quasi-Ehresmann semigroups. Our theorem generalizes and enriches the structure theorem given by M. Loganathan in 1987 for regular semigroups with a medial idempotent.

Differential and Analytical Properties of Semigroups of Operators

We systematically analyze differential and analytical properties of various kinds of semigroups of linear operators, including (local) convoluted C-semigroups and ultradistribution semigroups. The study of differentiable integrated semigroups leans heavily on the unification of the approaches of Barbu (Ann Scuola Norm Sup Pisa 23:413–429, 1969) and Pazy (Semigroups of linear operators and applications to partial differential equations. Springer, Berlin, 1983). We furnish illustrative examples of operators which generate differentiable integrated semigroups, further analyze the analytic properties of solutions of the backwards heat equation, and prove that several introduced classes of differentiable semigroups persist under bounded ‘commuting’ perturbations.

TYPE-A SEMIGROUPS WHOSE FULL SUBSEMIGROUPS FORM A CHAIN UNDER SET INCLUSION

A semigroup S is called a ∇-semigroup if the set of all its full subsemigroups forms a chain under set inclusion. In this paper, we investigate some properties of such kind of semigroups and establish several characterization theorems of type-A ∇-semigroups. Our theorems generalize the known results of P. R. Jones obtained in 1981 on inverse semigroups whose full inverse subsemigroups form a chain.

Generalized Bicyclic Semigroups and Jones Semigroups

In this paper, we consider the generalized bicyclic semigroups B n = 〈a, b | a n b = 1〉 and the Jones semigroups A n = 〈a, b | an+1b = a〉. They are the generalizations of the bicyclic semigroup B = 〈a, b | ab = 1〉 and its analogous semigroup A = 〈a, b | a 2 b = a〉 discovered by P.R., Jones in 1987. The word problem for these kinds of semigroups is solved. It is proved that, for n ≥ 2, B n are bisimple right inverse but not inverse semigroups and that the semigroup C = 〈a, b | a2b = a, ab2 = b〉 is the smallest idempotent-free homomorphic image of A n . Moreover, we also prove that A n and A m are mutually embeddable but not isomorphic with each other if n ≠ m. As a consequence, different kind of $${\cal D}$$ -nontrivial [0-]simple semigroups without idempotents are discussed.

On the Congruence Extension Property for Semigroups: Preservation under Homomorphic Images

A congruence ρT on a subsemigroup T of S extends to the semigroup S, if there exists a congruence ρ on S such that ρ∣T=ρT. A semigroup S has the congruence extension property (CEP) if each congruence on each subsemigroup extends to S. Previously, it was shown that a semigroup S has CEP if and only if it satisfies the so-called CEP conditions, a construction for semigroups with CEP was given, and we characterized congruences on this kind of semigroup. In the 1970s, the question whether or not CEP for semigroups is preserved in homomorphic images was asked. We give a positive answer for the problem in this paper.

Representation of congruences on regular semigroups with inverse transversals

For a regular semigroup with an inverse transversal, we have Saito’s structureW(I,S o, Λ, *, {α, β}). We represent congruences on this kind of semigroups by the so-called congruence assemblage which consist of congruences on the structure component partsI,S o and Λ. The structure of images of this type of semigroups is also presented.

Congruences on semigroups with the congruence extension property

A congruence ρT on a subsemigroup T of S extends to the semigroup S, if there exists a congruence ρ on S such that the restriction ρ |T of ρ on T is ρT. A semigroup S has the congruence extension property, CEP, if each congruence on each subsemigroup extends to S. Let S be a semigroup. Then S is called a GV-semigroup if S is π-regular and every regular element of S belongs to a subgroup(see[2]). For a congruence ρ on a semigroup S, the kernel kerρ of ρ is defined by {a ϵ S | a ρ e for some e ϵ E(SD)} and the trace trρ of ρ is defined by ρ | E(S). If S has CEP then by [14] S is a GV-semigroup. Thus we actually have kerρ = {a ϵ S | a ρ a 0}. It is known in [2] that a GV-semigroup S is a semilattice of nil-extensions of completely simple semigroups and the least semilattice congruence on S is . It is shown in [14] that a semigroup with CEP is a semilattice of nil-extensions of rectangular groups. The construction of semigroups with C'EP is given in [15]. In this paper, we first investigate congruences on several kinds of semigroups,such as nil-semigroups. π-groups and semigroups which are nil-extensions of rectangular groups. Then, we characterize congruences on a semigroup S = ∪{S α | α ϵ Y} with CEP by a congruence aggregate approach. It is shown that a ρ b if and only if a 0 trρ b0 and ab -1 ϵ kerρ for a, b ϵ (RegS)ρ. Since D * ϵ C(S), we obtain the congruence pair (ρ ⋂ D *, ρ ⋁ D *), which is an analogue of (ρ ⋂ D, ρ ⋁ D) by Petrich [12]. In a sequel, we prove that a congruence ρ on a semigroup with CEP is determined uniquely by the triple (η, π, δ). In addition, we characterize congruences on semigroups with CEP by a kernel-trace approach. This method for congruences on semigroups is similar to Pet,rich [12] and [13].