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Register a new userThe lattice of varieties of monoids
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We survey results devoted to the lattice of varieties of monoids. Along with known results, some unpublished results are given with proofs. A number of open questions and problems are also formulated.
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A variety is finitely based if it has a finite basis of identities. A minimal non-finitely based variety is called limit. A monoid is aperiodic if all its subgoups are trivial. Limit varieties of aperiodic monoids have been studied by Jackson, Lee, Zhang and Luo, Gusev and Sapir. In particular, Gusev and Sapir have recently reduced the problem of classifying all limit varieties of aperiodic monoids to the two tasks. One of them is to classify limit varieties of monoids satisfying $xsxt approx xsxtx$. In this paper, we completely solve this task. In particular, we exhibit the first example of a limit variety of monoids with countably infinitely many subvarieties. In view of the result by Jackson and Lee, the smallest known monoid generating a variety with continuum many subvarieties is of order six. It follows from the result by Edmunds et al. that if there exists a smaller example, then up to isomorphism and anti-isomorphism, it must be a unique monoid $P_2^1$ of order five. Our main result implies that the variety generated by $P_2^1$ contains only finitely based subvarieties and so has only countably many of them.
The set of all cancellable elements of the lattice of semigroup varieties has recently been shown to be countably infinite. But the description of all cancellable elements of the lattice $mathbb{MON}$ of monoid varieties remains unknown. This problem is addressed in the present article. The first example of a monoid variety with modular but non-distributive subvariety lattice is first exhibited. Then a necessary condition of the modularity of an element in $mathbb{MON}$ is established. These results play a crucial role in the complete description of all cancellable elements of the lattice $mathbb{MON}$. It turns out that there are precisely five such elements.
In the general context of presentations of monoids, we study normalisation processes that are determined by their restriction to length-two words. Garsides greedy normal forms and quadratic convergent rewriting systems, in particular those associated with the plactic monoids, are typical examples. Having introduced a parameter, called the class and measuring the complexity of the normalisation of length-three words, we analyse the normalisation of longer words and describe a number of possible behaviours. We fully axiomatise normalisations of class (4, 3), show the convergence of the associated rewriting systems, and characterise those deriving from a Garside family.
A limit variety is a variety that is minimal with respect to being non-finitely based. The two limit varieties of Marcel Jackson are the only known examples of limit varieties of aperiodic monoids. Our previous work had shown that there exists a limit subvariety of aperiodic monoids that is different from Marcel Jacksons limit varieties. In this paper, we introduce a new limit variety of aperiodic monoids.
We prove a sufficient condition under which a semigroup admits no finite identity basis. As an application, it is shown that the identities of the Kauffman monoid $mathcal{K}_n$ are nonfinitely based for each $nge 3$. This result holds also for the case when $mathcal{K}_n$ is considered as an involution semigroup under either of its natural involutions.
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