Definability in the lattice of equational theories of semigroups

We study first-order definability in the latticeL of equational theories of semigroups. A large collection of individual theories and some interesting sets of theories are definable inL. As examples, ifT is either the equational theory of a finite semigroup or a finitely axiomatizable locally finite theory, then the set {T, T ^ϖ} is definable, whereT ^ϖ is the dual theory obtained by inverting the order of occurences of letters in the words. Moreover, the set of locally finite theories, the set of finitely axiomatizable theories, and the set of theories of finite semigroups are all definable. The research of both authors was supported by National Science Foundation Grant No. DMS-8302295     

Definability for equational theories of commutative groupoids

We find several large classes of equations with the property that every automorphism of the lattice of equational theories of commutative groupoids fixes any equational theory generated by such equations, and every equational theory generated by finitely many such equations is a definable element of the lattice. We conjecture that the lattice has no non-identical automorphisms.     

Modular and lower-modular elements of lattices of semigroup varieties

The paper contains three main results. First, we show that if a commutative semigroup variety is a modular element of the lattice Com of all commutative semigroup varieties then it is either the variety $\mathcal{COM}$ of all commutative semigroups or a nilvariety or the join of a nilvariety with the variety of semilattices. Second, we prove that if a commutative nilvariety is a modular element of Com then it may be given within $\mathcal{COM}$ by 0-reduced and substitutive identities only. Third, we completely classify all lower-modular elements of Com. As a corollary, we prove that an element of Com is modular whenever it is lower-modular. All these results are precise analogues of results concerning modular and lower-modular elements of the lattice of all semigroup varieties obtained earlier by Je?ek, McKenzie, Vernikov, and the author. As an application of a technique developed in this paper, we provide new proofs of the ??prototypes?? of the first and the third our results.     

Bounded BCK‐algebras and their generated variety

In this paper we prove that the equational class generated by bounded BCK‐algebras is the variety generated by the class of finite simple bounded BCK‐algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK‐algebras is also a relatively simple bounded BCK‐algebra. Moreover, we show that every simple bounded BCK‐algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main results to some richer subreducts of the class of integral commutative bounded residuated lattices and to the involutive case. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Structural diversity in the lattice of equational theories

This paper is principally concerned with conditions under which various partition lattices are isomorphic to intervals in either the lattice of equational theories extending a given equational theory or the lattice of subtheories of a given equational theory. This paper is for Elizabeth Eldridge. Presented by W. Taylor.     

Commutative medial ternary groupoids

The paper studies the class of commutative medial ternary groupoids. A construction of ternary semiterms is given and it is proved that the equational theory of medial commutative ternary groupoids is solvable, namely, an algorithm is found, which in allmedial commutative ternary groupoids verifies the validity of the identity u = v for any pair (u, v) of terms. A construction of free medial commutative ternary groupoids is given, and it is proved that anymedial commutative ternary groupoid has a convex linear representation.     

Semigroups with Boolean congruence lattices

A construction of all globally idempotent semigroups with Boolean (complemented modular, relatively complemented, sectionally complemented, respectively) congruence lattice is given. Furthermore, it is shown that an arbitrary semigroup has Boolean (...) congruence lattice if and only if it is a special kind of inflation of a semigroup of the foregoing type. As applications, all commutative, finite, and completely semisimple semigroups, respectively, with Boolean (...) congruence lattice are completely determined.     

交换矩阵半群的可约性

首先分别给出单生矩阵半群或者摹群不可约、不可分解以及完全可约的充分必要条件,其次讨论一般域上矩阵半群的可约性的一些条件,最后特别地讨论实数域上矩阵半群的可约性,完全确定了实数域上对称和反对称矩阵组成的不可约交换矩阵半群.     

Note on exclusive semigroups

In the SEMIGROUP FORUM, Vol. 1, No. 1, B. M. Schein proposed the following problem: Describe the structure of semigroups S such that for every a,b,c∈S, abc=ab, bc or ac. At present, we shall call such a semigroup S anexclusive semigroup. Recently, the author heard that the structure of commutative exclusive semigroups was completely determined by T. Tamura [3]. In this paper, we deal with exclusive semigroups which are not necessarily commutative. The paper is divided into three sections. At first, the structure of exclusive semigroups whose idempotents form a rectangular band will be clarified. Next, we shall investigate a certain class of exclusive semigroups called “exclusive homobands”. Especially, in the final section we shall deal with medial exclusive homobands and show how to construct them. The proofs are omitted and will be given in detail elsewhere.     

Some subdirectly irreducible idempotent semigroups

Subdirectly irreducible idempotent semigroups were characterized in [3], and in that paper, their connection with the various equational classes of idempotent semigroups was discussed. All these results are in terms of identities, so that examples of subdirectly irreducibles in the equational classes are explicitly known only for small classes. It is easy to show from general considerations (see the last section of the present paper) that every proper equational subclass of the class of idempotent semigroups is generated (as an equational class) by one or two subdirectly irreducibles. In this paper we give an example of a subdirectly irreducible for each join irreducible equational class of idempotent semigroups, which generates the class. This list, together with known results, gives explicit examples of one or two finite subdirectly irreducibles which generate the various equational classes. Research supported by the National Research Council of Canada.     

Lattices of equational theories are congruence lattices of monoids with one additional unary operation

The problem of characterizing the lattices of equational theories is still unsolved. In this paper it is shown that these lattices are congruence lattices of an algebra whose fundamental operations consist of one monoid operation with right zero and one unary operation.Presented by R. McKenzie.     

Lower-Modular Elements of the Lattice of Semigroup Varieties

We call a semigroup variety modular [upper-modular, lower-modular, neutral] if it is a modular [respectively upper-modular, lower-modular, neutral] element of the lattice of all semigroup varieties. It is proved that if V is a lower-modular variety then either V coincides with the variety of all semigroups or V is periodic and the greatest nil-subvariety of V may be given by 0-reduced identities only. We completely determine all commutative lower-modular varieties. In particular, it turns out that a commutative variety is lower-modular if and only if it is neutral. A number of corollaries of these results are obtained.     

Varieties and Nilpotent Extensions of Permutative Periodic Semigroups

We investigate certain semigroup varieties formed by nilpotent extensions of orthodox normal bands of commutative periodic groups. Such semigroups are shown to be both structurally periodic and structurally commutative, and are therefore structurally inverse semigroups. Such semigroups are also shown to be dense semilattices of structurally group semigroups. Making use of these structure decompositions, we prove that the subvariety lattice of any variety comprised of such semigroups is isomorphic to the direct product of the following three sublattices: its sublattice of all structurally trivial semigroup varieties, its sublattice of all semilattice varieties, and its sublattice of all group varieties. We conclude, therefore, that to completely describe this lattice, we must first describe completely the lattice of all structurally trivial semigroup varieties, since the other two are well known lattices.     

On definability of types and relative stability

In this paper, we consider the question of definability of types in non‐stable theories. In order to do this we introduce a notion of a relatively stable theory: a theory is stable up to Δ if any Δ‐type over a model has few extensions up to complete types. We prove that an n‐type over a model of a theory that is stable up to Δ is definable if and only if its Δ‐part is definable.     

关于弱交换PO—半群中的一个问题

本文引入弱交换ｐｏ－半群的概论２，研究这类半群到Ａｒｃｈｉｍｅｄｅａｎ子半群的半格分解，得到了这半群类似于具平凡序的弱交换半群的一个特征，由此在更一般的情形下回答了Ｋｅｈａｙｏｐｕｌｕ在「１」中提出的一个问题，并作为推论得到弱交换ｐｏｅ－半群和具平凡序的弱交换半群的已知结果。     

由格蕴涵代数诱导的伴随半群

给出由格蕴涵代数诱导出的伴随半群及有关概念 ,详细讨论伴随半群中的元素即格蕴涵代数的左映射的性质 ,得到它们的几个等价条件。最后讨论由格蕴涵代数诱导的两个双格半群与伴随半群之间的关系 ,并证明这些半群是幂等的当且仅当它们是由格 H蕴涵代数所诱导     

Commutative semigroups whose homomorphic images in groups are groups

In this paper we study commutative semigroups whose every homomorphic image in a group is a group. We find that for a commutative semigroup S, this property is equivalent to S being a union of subsemigroups each of which either has a kernel or else is isomorphic to one of a sequence T₀, T₁, T₂, ... of explicitly given, countably infinite semigroups without idempotents. Moreover, if S is also finitely generated then this property is equivalent to S having a kernel.     

交换半群的强半格

本文刻划交换半群的强半格上的最小半格同余，并证明由此得到的商半群为对应的每个交换半群的商半群的强半格。     

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.