Reducibility of Joins Involving Some Locally Trivial Pseudovarieties

Abstract

In this paper, we show that σ-reducibility is preserved under joins with K, where K is the pseudovariety of semigroups in which idempotents are left zeros. Reducibility of joins with D, the pseudovariety of semigroups in which idempotents are right zeros, is also considered. In this case, we were able to prove that σ- reducibility is preserved for joins with pseudovarieties verifying a certain property of cancellation. As an example involving the semidirect product, we prove that Sl*K is κ-tame, where Sl stands for the pseudovariety of semilattices.     

Tameness of Pseudovariety Joins Involving ${\sf R}$

In this paper, we establish several decidability results for pseudovariety joins of the form , where is a subpseudovariety of or the pseudovariety . Here, (resp. ) denotes the pseudovariety of all -trivial (resp. -trivial) semigroups. In particular, we show that the pseudovariety is (completely) κ-tame when is a subpseudovariety of with decidable κ-word problem and is (completely) κ-tame. Moreover, if is a κ-tame pseudovariety which satisfies the pseudoidentity x₁ ⋯ x_ry^ω+1zt^ω = x₁ ⋯ x_ryzt^ω, then we prove that is also κ-tame. In particular the joins , , , and are decidable. Partial support by FCT, through the Centro de Matemática da Universidade do Porto, is also gratefully acknowledged. Partial support by FCT, through the Centro de Matemática da Universidade do Minho, is also gratefully acknowledged.     

Varieties of Structurally Trivial Semigroups II

Abstract. Melnik [5] determined completely the lattice of all 2-nilpotent extensions of rectangular band varieties; and Koselev [4] determined a distributive sublattice formed by certain varieties of n-nilpotent extensions of left zero bands. In [2] the author described the skeleton of the lattice of all 3-nilpotent extensions of rectangular bands. We generalize these results by proving that a certain family of semigroup varieties which includes all the varieties mentioned above, and referred to here as planar varieties, consisting of certain n-nilpotent extensions of rectangular bands forms a distributive sublattice that looks somewhat like an inverted pyramid. Our proof makes use of a countably infinite family of injective endomorphisms on the lattice of all semigroup varieties that was introduced by the author in [1]. Although we do not determine completely the lattice of all n-nilpotent extensions of rectangular band varieties, our result unifies certain previously known results and provides a framework for further research.     

Varieties of Structurally Trivial Semigroups III

The skeleton of the lattice of all structurally trivial semigroup varieties is known to be isomorphic to an infinitely ascending inverted pyramid (Kopamu, 2003 Kopamu , S. J. L. ( 2003 ). Varieties of structurally trivial semigroups II . Semigroup Forum 66 : 401 – 415 . [Google Scholar]). We digitize the skeleton by representing each variety forming the skeleton as an ordered triple of non-negative integers. This digitization of the lattice, under the pointwise ordering of non-negative integers, provides useful algorithms which could easily be programmed into a computer, and then used to compute varietal joins and meets, or even to draw skeleton lattice diagrams. An application to a certain larger subvariety lattice is also given as an example.     

群伪簇与幂零半群伪簇的两种积

本文确定了群伪簇Ｇ与幂零半群伪簇的Ｎ的半直积Ｇ＊Ｎ,Ｎ＊Ｇ和Ｍａｌｃｅｖ积Ｇ＾－１Ｎ,Ｎ＾－１Ｇ。     

Zero-Divisor Semigroups and Some Simple Graphs

In this article, we study commutative zero-divisor semigroups determined by graphs. We prove that for all n ≥ 4, the complete graph K _n together with two end vertices has a unique corresponding zero-divisor semigroup, while the complete graph K _n together with three end vertices has no corresponding semigroups. We determine all the twenty zero-divisor semigroups whose zero-divisor graphs are the complete graph K ₃ together with an end vertex.     

Some Remarks on Additive Arithmetical Semigroups

Under weak conditions called axioms and , we prove Chebyshevtype estimates and asymptotic formulas, respectively, for the prime elements in general additive arithmetical semigroups. As applications, we derive asymptotic laws for the mean behavior of prime divisor functions and of distinct degrees of prime factors.     

序半群的若干等价性质

本主要建立了序半群S中N-类的单性与S的关于其元素和理想的某些性质之间的等介关系。     

Some Normal Cryptogroups as Semigroups of Isomorphisms

Given a Clifford semigroup G, we construct special G-operands L and R which we term conformai. Certain suboperands of L and R we call threads and fix some special G-isomorphisms, which we term coherent, of threads in R onto threads in L. On the set of all coherent G-isomorphisms of threads in L onto threads in R we define a sandwich-type multiplication. When we restrict our threads to be cyclic suboperands of L and R, this construction produces a normal cryptogroup which we represent as $ S=[Y;S_{\alpha},\chi_{{\alpha},{\beta}}] $ -Without any restriction on the threads this produces a semigroup isomorphic with a remarkable ideal of the translational hull of S. Conversely, given a strong semilattice of completely simple semigroups, satisfying certain conditions, we can represent it isomorphically as indicated above.     

Some Characterizations of Strongly Semisimple Ordered Semigroups

In this paper, the concept of quasi-prime fuzzy left ideals of an ordered semigroup $S$ is introduced. Some characterizations of strongly semisimple ordered semigroups are given by quasi-prime fuzzy left ideals of $S$. In particular, we prove that $S$ is strongly semisimple if and only if each fuzzy left ideal of $S$ is the intersection of all quasi-prime fuzzy left ideals of $S$ containing it.     

Some Constructions Related to Rees Matrix Semigroups

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On Some Semigroups on the Complex Plane

It is shown that the sets $C(\alpha)=\left\{z\in\dC: |z\sin\alpha\pm i\cos \alpha|\le 1\right\}$, where $\alpha\in (0,\pi/2)$ form multiplicative semigroups on the complex plane $\dC$. We prove that the semigroups $C(\alpha)$ and $C(\beta)$ are not isomorphic when $\alpha\ne \beta$ and the unique automorphisms of the semigroup $C(\alpha)$ are the mappings $\Phi(z)=z$ and $\Phi(z)=\overline z$. All continuous semicharacters of the semigroups $C(\alpha)$ and all continuous automorphisms of the closed unit disk are described. Other examples of semigroups on the complex are obtained by transformations of $C(\alpha).$     

The Power Exponent of a Pseudovariety

\noindent A characterization of the pseudovarieties of finite semigroups whose powers are contained in \D S is given. This characterization, combined with some known results, classifies all pseudovarieties in terms of their power exponent.     

Trivial automorphisms

We prove that the statement ‘For all Borel ideals I and J on ω, every isomorphism between Boolean algebras P(ω)/I and P(ω)/J has a continuous representation’ is relatively consistent with ZFC. In this model every isomorphism between P(ω)/I and any other quotient P(ω)/J over a Borel ideal is trivial for a number of Borel ideals I on ω. We can also assure that the dominating number, σ, is equal to ?₁ and that \({2^{{\aleph _1}}} > {2^{{\aleph _0}}}\) . Therefore, the Calkin algebra has outer automorphisms while all automorphisms of P(ω)/Fin are trivial. Proofs rely on delicate analysis of names for reals in a countable support iteration of Suslin proper forcings.     

具有某种断面的半群的研究进展

本文综述了几类具有特殊断面的半群的近期研究结果。在介绍逆半群和正则半群的一般结构之后，概述了具有逆断面的正则半群的结构和同余格的研究成果。总结了作为逆断面的推广的可裂断面，纯正断面，正则^*－断面和恰当断面。提出了可以进一步研究的重要的问题。     

阿基米德序半群的链的若干刻画

本文首先引入了一个序半群$S$的准素模糊理想的概念,通过序半群$S$上的一些二元关系以及它的理想的模糊根给出了该序半群是阿基米德序子半群的半格的一些刻画.进一步地借助于序半群$S$的模糊子集对该序半群是阿基米德序子半群的半格进行了刻画.尤其是通过序半群的模糊素根定理证明了序半群$S$是阿基米德序子半群的链当且仅当$S$是阿基米德序子半群的半格且$S$的所有弱完全素模糊理想关于模糊集的包含关系构成链.     

Some Characterizations of Left Weakly Regular Ordered Semigroups

In this paper, the notion of left weakly regular ordered semigroups is introduced. Furthermore, left weakly regular ordered semigroups are characterized by the properties of their left ideals, right ideals and (generalized) bi-ideals, and also by the properties of their fuzzy left ideals, fuzzy right ideals and fuzzy (generalized) bi-ideals.