Lattices of semigroup varieties

We survey a great number of results obtained during four decades of investigations on lattices of semigroup varieties and formulate several open problems.     

Finitely generated permutative varieties

We show that there exists an algorithm which decides whether a finite set of identities containing a permutation identity defines a variety generated by a finite semigroup or not.     

Varieties of restriction semigroups and varieties of categories

The variety of restriction semigroups may be most simply described as that generated from inverse semigroups (S, ·, ^?1) by forgetting the inverse operation and retaining the two operations x⁺ = xx^?1 and x* = x^?1x. The subvariety B of strict restriction semigroups is that generated by the Brandt semigroups. At the top of its lattice of subvarieties are the two intervals [B₂, B₂M = B] and [B₀, B₀M]. Here, B₂ and B₀ are, respectively, generated by the five-element Brandt semigroup and that obtained by removing one of its nonidempotents. The other two varieties are their joins with the variety of all monoids. It is shown here that the interval [B₂, B] is isomorphic to the lattice of varieties of categories, as introduced by Tilson in a seminal paper on this topic. Important concepts, such as the local and global varieties associated with monoids, are readily identified under this isomorphism. Two of Tilson's major theorems have natural interpretations and application to the interval [B₂, B] and, with modification, to the interval [B₀, B₀M] that lies below it. Further exploration may lead to applications in the reverse direction.     

All Reg-solid varieties of commutative semigroups

We determine all regular solid varieties of commutative semigroups. Each of them is contained in the Reg-hyperequational class V _RC defined by the associative law and the commutative law, and every subvariety of V _RC is regular solid. In the present paper, the subvariety lattice of V _RC will be characterized.     

Every numerical semigroup is one half of a symmetric numerical semigroup

Let be a numerical semigroup. Then there exists a symmetric numerical semigroup such that .

     

A Ring Variety without an Independent Basis

An example of a ring variety without an independent basis is constructed. It is proved that this variety is the intersection of two independently based varieties.     

半群的i-v Fuzzy理想

在半群上引入i-v Fuzzy理想的概念.研究了半群的i-v Fuzzy理想的若干性质,特别是给出了半群的i-v Fuzzy子集成为理想的若干特征性质.     

The lattice of semigroup varieties

Semigroups are considered here in terms of the identities which they may satisfy with the emphasis on the identities and on classes of semigroups defined by identities, rather than on the semigroups themselves. These classes, called varieties, form a lattice under inclusion and it is illuminating to interpret properties of semigroup identities and varieties in terms of this lattice, its cardinality, its atoms, its infinite chains, and so on. This paper is an expansion of addresses to the Southeastern Section of the Mathematical Association of America at East Carolina University, March 1968 and to the Symposium on Semigroups and Rings at the University of Puerto Rico, Mayaguez, March 1970. The preparation of the paper was supported in part by NSF Grants GP 6597 and GP 20638.     

Approximate amenability of certain inverse semigroup algebras

In this article, the approximate amenability of semigroup algebra ?¹(S) is investigated, where (S) is a uniformly locally finite inverse semigroup. Indeed, we show that for a uniformly locally finite inverse semigroup (S), the notions of amenability, approximate amenability and bounded approximate amenability of ?¹(S) are equivalent. We use this to give a direct proof of the approximate amenability of ?¹(S) for a Brandt semigroup (S). Moreover, we characterize the approximate amenability of ?¹(S), where S is a uniformly locally finite band semigroup.     

Dense inverse subsemigroups of a topological inverse semigroup

In this paper we study dense inverse subsemigroups of topological inverse semigroups. We construct a topological inverse semigroup from a semilattice. Finally, we give two examples of the closure of B _{( −∞, ∞ )}¹, a topological inverse semigroup obtained by starting with the real numbers as a semilattice with the operation a ∨ b=sup{a,b}. The author would like to thank to the referee for useful suggestions.     

Continuous extension of a densely parameterized semigroup

Let be a dense sub-semigroup of ℝ₊, and let X be a separable, reflexive Banach space. This note contains a proof that every weakly continuous contractive semigroup of operators on X over can be extended to a weakly continuous semigroup over ℝ₊. We obtain similar results for nonlinear, nonexpansive semigroups as well. As a corollary we characterize all densely parametrized semigroups which are extendable to semigroups over ℝ₊. O.M. Shalit was partially supported by the Gutwirth Fellowship.     

The arithmetic extensions of a numerical semigroup

Abstract

In this paper we introduce the notion of extension of a numerical semigroup. We provide a characterization of the numerical semigroups whose extensions are all arithmetic and we give an algorithm for the computation of the whole set of arithmetic extension of a given numerical semigroup. As by-product, new explicit formulas for the Frobenius number and the genus of proportionally modular semigroups are obtained.     

The convergence of Lax-Oleinik semigroup for time-periodic Lagrangian

In this article,the convergence of so-called Lax-Oleinik semigroup is studied for time-periodic Lagrangian systems when the degree of freedom is greater than 2.Under certain conditions,we show that the Lax-Oleinik semigroup converges if the rotation vector is completely irrational.Removing such conditions,we will give another kind of convergence of the sequence Fc((x,s),(x′,s′+Tn)),the convergence of which is closely related to the Lax-Oleinik semigroup.     

The universal nilpotent group compactification of a semigroup

The purpose of this paper is to introduce an algebra of functions on a semitopological semigroup and to study these functions from the point of view of universal semigroup compactification. We show that the corresponding semigroup compactification of this algebra is universal with respect to the property of being a nilpotent group.

     

The rank of a commutative cancellative semigroup

Summary For a commutative cancellative semigroup S, we define the rank of S intrinsically. This definition implies that the rank of S equals the usual rank of its group of quotients. We also characterize the rank in terms of embeddability into a rational vector space of the greatest power cancellative image of S.     

幂等元位于中心的半群的局部化和最小幂幺半群同余

局部化是交换代数的重要工具[1],证明幂等元位于中心的半群在其幂等元半格上的局部化存在且唯一,并给出此类半群的最小幂幺半群同余.另外,给出了若干半群的重要同余的刻划.     

Balanced subsets of a symmetric inverse semigroup

For elements in the subsets of some symmetric inverse semigroup we study the problem of equal cardinality for the sets of commuting and noncommuting elements.