All maximal unit-regular elements of Relhyp((m),(n))

Abstract

Any relational hypersubstitution for algebraic systems of type (τ, τ′) = ((mi)i∈I , (nj )j∈J ) is a mapping which maps any mi-ary operation symbol to an mi-ary term and maps any nj-ary relational symbol to an nj-ary relational term preserving arities, where I, J are indexed sets. The set of all relational hypersubstitutions for algebraic systems of type (τ, τ′) together with a binary operation defined on the set and its identity forms a monoid. The properties of this structure are expressed by terms and formulas. Some algebraic properties of the monoid of a special type, especially the set of all unit-regular elements, were studied. In this paper, we determine all maximal unit-regular submonoids of this monoid of type ((m), (n)) for arbitrary natural numbers m, n ≥ 2.