Hereditarily finitely based monoids of extensive transformations

It is shown that the monoid E _n of extensive transformations of a chain of order n is hereditarily finitely based if and only if n ≤ 3. It follows that the submonoid OE _n of order-preserving transformations in E _n is also hereditarily finitely based if and only if n ≤ 3.     

Finite basis problem for 2-testable monoids

A monoid S ¹ obtained by adjoining a unit element to a 2-testable semigroup S is said to be 2-testable. It is shown that a 2-testable monoid S ¹ is either inherently non-finitely based or hereditarily finitely based, depending on whether or not the variety generated by the semigroup S contains the Brandt semigroup of order five. Consequently, it is decidable in quadratic time if a finite 2-testable monoid is finitely based.     

Enumeration of symplectic and orthogonal injective partial transformations

In this paper, we compute the generating function of , where a is a real number with a≥1. We then use this function to determine the generating functions of the symplectic and orthogonal Renner monoids. Furthermore, we show that these functions are closely related to Laguerre polynomials.     

On the divisibility problem for one-relator monoids

This work was supported by the Russian Foundation for Fundamental Research (Under Grant No. 93-011-16015).     

On the finite basis problem for certain 2-limited words

Let X* be a free monoid over an alphabet X and W be a finite language over X. Let S(W) be the Rees quotient X*/I(W), where I(W) is the ideal of X* consisting of all elements of X* that are not subwords of W. Then S(W) is a finite monoid with zero and is called the discrete syntactic monoid of W. W is called finitely based if the monoid S(W) is finitely based. In this paper, we give some sufficient conditions for a monoid to be non-finitely based. Using these conditions and other results, we describe all finitely based 2-limited words over a three-element alphabet. Furthermore, an explicit algorithm is given to decide that whether or not a 2-limited word in which there are exactly two non-linear letters is finitely based.     

On a finite basis problem for universal positive formulas

A finite semigroup without a finite basis of collective identities has been constructed. A semigroup with a finite basis of identities, but without a finite basis of disjunctive identities has been constructed.Presented by B. M. Schein.     

On Schützenberger monoids in transformations semigroups

K. D. Magill and S. Subbiah [4] have studied the semigroup of continuous selfmaps of a topological space. They have shown that the Schützenberger group of theH containing a selfmap a is in a close relation to the group of homeomorphic selfmaps of the range of a. There arose a naturalproblem to find a generalization of the Schützenberger group, which is in the similar relation to the semigroup of continuous selfmaps of the range of a. C. E. Clark and J. H. Carruth [1] introduced a notion of a Schützenberger monoid. In this paper Schützenberger monoids in an arbitrary semigroup of transformations are studied and it is shown that this notion is the generalization we were looking for.     

Partial automorphisms and injective partial endomorphisms of a finite undirected path

Semigroup Forum - In this paper, we study partial automorphisms and, more generally, injective partial endomorphisms of a finite undirected path from Semigroup Theory perspective. Our main...     

On the word problem for syntactic monoids of piecewise testable languages

Piecewise testable languages are widely studied area in the theory of automata. We analyze the algebraic properties of these languages via their syntactic monoids. In this paper a normal form is presented for 2- and 3-piecewise testable languages and a log-asymptotic estimate is given for the number of words over these monoids.     

On free spectra of finite monoids from the pseudovariety DA

The pseudovariety DA consists of all aperiodic finite monoids all of whose regular -classes are subsemigroups (that is, rectangular subbands); this pseudovariety appears quite frequently in various contexts in finite semigroup theory. In this note we prove that all its members have a log-polynomial free spectrum, thereby making a new step towards proving the Seif conjecture on the dichotomy of free spectra of finite monoids.     

Maximal partial clones with no finite basis

Let A be a nonsingleton finite set. We give a criterion for recognizing not finitely generated strong partial clones and show that none of the |A| strong maximal partial clones of Słupecki type over A has a finite basis. Received November 14, 1997; accepted in final form August 18, 1998.