Prakit Jampachon

REGULAR ELEMENTS AND GREEN'S RELATIONS IN MENGER ALGEBRAS OF TERMS

Defining an (n + 1)-ary superposition operation Sn on the set Wτ (Xn) of all n-ary terms of type τ , one obtains an algebra n − clone τ := (Wτ (Xn); Sn, x1, . . . , xn) of type (n + 1, 0, . . . , 0). The algebra n − clone τ is free in the variety of all Menger algebras ([9]). Using the operation Sn there are different possibilities to define binary associative operations on the set Wτ (Xn) and on the cartesian power Wτ (Xn) n. In this paper we study idempotent and regular elements as well as Green’s relations in semigroups of terms with these binary associative operations as fundamental operations.

Idempotent elements of weak projection generalized hypersubstitutions

Abstract A generalized hypersubstitution of type τ = (ni)i∈I is a mapping σ which maps every operation symbol fi to the term σ (fi) and may not preserve arity. It is the main tool to study strong hyperidentities that are used to classify varieties into collections called strong hypervarieties. Each generalized hypersubstitution can be extended to a mapping σ̂ on the set of all terms of type τ. A binary operation on HypG(τ), the set of all generalized hypersubstitutions of type τ, can be defined by using this extension. The set HypG(τ) together with such a binary operation forms a monoid, where a hypersubstitution σid, which maps fi to fi(x1, . . . , xn₁ ) for every i ∈ I, is the neutral element of this monoid. A weak projection generalized hypersubstitution of type τ is a generalized hypersubstitution of type τ which maps at least one of the operation symbols to a variable. In semigroup theory, the various types of its elements are widely considered. In this paper, we present the characterizations of idempotent weak projection generalized hypersubstitutions of type (m, n) and give some sufficient conditions for a weak projection generalized hypersubstitution of type (m, n) to be regular, where m, n ≥ 1.