The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category D. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott (D “ sets), the syntactic semirings of Polák (D “ semilattices), and the syntactic associative algebras of Reutenauer (D = vector spaces)....

Generalizing results of Otto & Ruškuc, this paper shows that every finitely generated submonoid of a monoid presented by a confluent finite special rewriting system admits an automatic structure that is simultaneously rr-, `r-, r`-, and ``-automatic; and that every finitely generated submonoid of a monoid presented by a confluent regular monadic rewriting system admits an automatic structure th...

in this paper we study the notions of cogenerator and subdirectlyirreducible in the category of s-poset. first we give somenecessary and sufficient conditions for a cogenerator $s$-posets.then we see that under some conditions, regular injectivityimplies generator and cogenerator. recalling birkhoff'srepresentation theorem for algebra, we study subdirectlyirreducible s-posets and give this theo...

We survey the known results on two old open problems on commutation of languages. The first problem, raised by Conway in 1971, is asking if the centralizer of a rational language must be rational as well – the centralizer of a language is the largest set of words commuting with that language. The second problem, proposed by Ratoandromanana in 1989, is asking for a characterization of those lang...

The word problem for inverse monoids generated by a set Γ subject to relations of the form e = f , where e and f are both idempotents in the free inverse monoid generated by Γ , is investigated. It is shown that for every fixed monoid of this form the word problem can be solved in polynomial time which solves an open problem of Margolis and Meakin. For the uniform word problem, where the presen...

This paper studies rr-, lr-, rl-, and ll-automatic structures for finitely generated submonoids of monoids presented by confluent rewriting system that are either finite and special or regular andmonadic. A new technique is developed that uses an automaton to ‘translate’ betweenwords in the original rewriting system andwords over the generators for the submonoid. This is applied to show that th...

Margolis, S.W. and J.E. Pin, New results on the conjecture of Rhodes and on the topological conjecture, Journal of Pure and Applied Algebra 80 (1992) 305-313. The Conjecture of Rhodes, originally called the ‘type II conjecture’ by Rhodes, gives an algorithm to compute the kernel of a finite semigroup. This conjecture has numerous important consequences and is one of the most attractive problems...

Abstract We describe how one can explicitly obtain all atoms of an arbitrary root-closed monoid, whose quotient group is isomorphic to $$\mathbb {Z}^2$$ Z2 . For this purpose, we solve task for three special types such monoids in Theorems 5 and 6, then ...

We generalize to eventually regular (or ‘π-regular’) semigroups the study of the lattice of full regular subsemigroups of a regular semigroup, which has its most complete exposition in the case of inverse semigroups. By means of a judicious definition, it is shown that the full eventually regular subsemigroups of such a semigroup form a complete lattice LF , which projects onto the lattices of ...

In Ufnarovski (1989), the concept of automaton algebras is introduced. These are quotients of the non-commutative polynomial ring where the defining ideal allows some Gröbner basis with a regular set of leading words. However, nothing is reflected concerning the whole structure of the Gröbner basis (except of course for monomial algebras). In this paper we introduce the concept of regular Gröbn...