Upper-modular elements of the lattice of semigroup varieties. II

A semigroup variety is called a variety of degree ≤2 if all its nilsemigroups are semigroups with zero multiplication, and a variety of degree >2 otherwise. We completely determine all semigroup varieties of degree >2 that are upper-modular elements of the lattice of all semigroup varieties and find quite a strong necessary condition for semigroup varieties of degree ≤2 to have the same property.     

Distributive and neutral elements of the lattice of commutative semigroup varieties

We completely describe commutative semigroup varieties that are distributive, standard, or neutral elements of the lattice of all commutative semigroup varieties. In particular, we prove that the properties of being a distributive element and of being a standard element in this lattice are equivalent.     

Upper-modular and related elements of the lattice of commutative semigroup varieties

We completely determine upper-modular, codistributive and costandard elements in the lattice of all commutative semigroup varieties. In particular, we prove that the properties of being upper-modular and codistributive elements in the mentioned lattice are equivalent. Moreover, in the nil-case the properties of being elements of all three types turn out to be equivalent.     

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.     

The subvariety lattice of the join of two semigroup varieties

The join A ∨ B of two semigroup varieties A and B is investigated. The latrine of subvarieties of A ∨ B is completely described, It is shown that this lattice is finite and non-modular and that all varieties in it are finitely based and finitely generated.     

Two operators on the lattice of completely regular semigroup varieties

In this paper, varieties of completely regular semigroups are studied. This paper is divided into six sections. Section 1 contains an introduction to varieties of completely regular semigroups and preliminaries. Most of the notation needed in this paper is given. In Section 2, the operators \La ( ) and \Ra ( ) on the lattice of subvarieties of varieties of completely regular semigroups are investigated. In Section 3, some further properties of the operators \La ( ) and \Ra ( ) are given. In Section 4, the semigroups generated by various subset of some operators are considered. In Section 5, the operators \La ( ) and \Ra ( ) are used in finding the join of two given varieties. The word problem for free objects in the variety OLBG is considered in Section 6 using the operator \La ( ) . June 1, 1999     

Identities and quasiidentities in the lattice of overcommutative semigroup varieties

We describe overcommutative varieties of semigroups whose lattice of overcommutative subvarieties satisfies a non-trivial identity or quasiidentity. These two properties turn out to be equivalent.     

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.     

Special elements of the lattice of monoid varieties

We completely classify all neutral and costandard elements in the lattice \(\mathbb {MON}\) of all monoid varieties. Further, we prove that an arbitrary upper-modular element of \(\mathbb {MON}\) except the variety of all monoids is either a completely regular or a commutative variety. Finally, we verify that all commutative varieties of monoids are codistributive elements of \(\mathbb {MON}\). Thus, the problems of describing codistributive or upper-modular elements of \(\mathbb {MON}\) are completely reduced to the completely regular case.     

Special elements of the lattice of epigroup varieties

We describe right-hand skew Boolean algebras in terms of a class of presheaves of sets over Boolean algebras called Boolean sets, and prove a duality theorem between Boolean sets and étalé spaces over Boolean spaces.     

Covering elements in the lattice op varieties of algebras

Let variety μ be given by the balanced identities of signature Ω not containing unary operations. Then, in the lattice of subvarieties of variety μ, any element different from μ has an element covering it. In particular, variety μ might be the varieties of semigroups, groupoids, n-associatives, etc. It is also proven that, in the lattice of varieties of semigroups, there exists an element having a continuum of covering elements.     

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.