On the Lattice of Overcommutative Varieties of Monoids

Abstract

It is unknown so far, whether the lattice of all varieties of monoids satisfies some non-trivial identity. The objective of this note is to give a negative answer to this question. Namely, we prove that any finite lattice is a homomorphic image of some sublattice of the lattice of overcommutative varieties of monoids (i.e., varieties that contain the variety of all commutative monoids). This implies that the lattice of overcommutative varieties of monoids, and therefore, the lattice of all varieties of monoids does not satisfy any non-trivial identity.We study the lattice of varieties of monoids, i.e., algebras with two operations, namely an associative binary operation and a 0-ary operation that fixes the neutral element. There are many articles devoted to varieties of monoids. But the most part of them deals with the identities of the monoids. At the same time, only a little information is known so far about the lattice of monoid varieties which we denote by MON. In the works [4] and [12], respectively, the lattices of all commutative and all idempotent monoid varieties are described. In [7, Theorem 1], an example of a monoid variety without covers in the lattice MON is discovered. In [5, Subsection 3.2] two monoid varieties are exhibited such that the subvariety lattices of both the varieties are finite, while the subvariety lattice of their join is uncountably infinite and does not satisfy the ascending chain condition. Also, there are a few papers where a description of subvariety lattices of some concrete monoid varieties appeared as auxiliary results (see [6, Lemma 4.1], for instance). These results probably form a majority of all currently known results regarding the lattice MON. This is in sharp contrast with a large number of striking and deep results about the lattice of semigroup varieties obtained so far (see the survey [10]).Due to insufficient knowledge of the lattice of monoid varieties, many natural questions here remain open so far. In particular, it is unknown so far, whether this lattice satisfies some non-trivial identity. The objective of this note is to give a negative answer to this question. Note that the negative answer to the analogous question for the lattice of semigroup varieties was found in 1971 in two articles by Burris and Nelson [1,2].A variety of monoids is called overcommutative if it contains the variety of all commutative monoids. Evidently, the class of all overcommutative varieties forms a sublattice in the lattice of all monoid varieties, and we denote this sublattice by OC. The main result of this article is the following 2010 Mathematics Subject Classification. Primary 20M07, secondary 08B15.

Varieties of monoids with complex lattices of subvarieties

“…Since the class of all finite lattices violates every non-trivial quasi-identity [1], the interval [COM, O] also violates every non-trivial quasi-identity. This generalizes the aforementioned result of Gusev [5] about the interval [COM, MON]. More generally, for any finitely universal variety V, the lattice L(V) violates every non-trivial quasi-identity.…”

“…Overcommutative varieties constitute the interval [COM, MON]. Recently, Gusev [5] proved that the interval [COM, MON] violates every non-trivial lattice identity; the complexness of this interval naturally led to the following question. An affirmative answer to Question 1.2 clearly implies an affirmative answer to Question 1.1.…”

“…Gusev [5, Question]). Does the interval [COM, MON] contain an isomorphic copy of every finite lattice?…”

Special elements of the lattice of monoid varieties

“…Thus, Y ⊂ X. It is well known and can be easily verified that if a monoid variety does not contain C n+1 then this variety satisfies the identity (2) x n ≈ x n+m for some natural m (see [5,Lemma 2.5], for instance). In particular, an identity of such a form holds in V. Then V violates the identity…”

Varieties of monoids with complex lattices of subvarieties

“…Overcommutative varieties constitute the interval [COM, MON]. Recently, Gusev [5] proved that the interval [COM, MON] violates every nontrivial lattice identity; the complexness of this interval naturally led to the following question.…”