Unsolid and fluid strong varieties of partial algebras (Q933857)

Summary: A partial algebra \({\mathcal A}= (A;(f_i^A)_{i\in I})\) consists of a set \(A\) and an indexed set \((f_i^A)_{i\in I}\) of partial operations \(f_i^A: A^{n_i}{\multimap\to} A\). Partial operations occur in the algebraic description of partial recursive functions and Turing machines. A pair of terms \(p\approx q\) over the partial algebra \({\mathcal A}\) is said to be a strong identity in \({\mathcal A}\) if the right-hand side is defined whenever the left-hand side is defined and vice versa, and both are equal. A strong identity \(p\approx q\) is called a strong hyperidentity if when the operation symbols occurring in \(p\) and \(q\) are replaced by terms of the same arity, the identity which arises is satisfied as a strong identity. If every strong identity in a strong variety of partial algebras is satisfied as a strong hyperidentity, the strong variety is called solid. In this paper, we consider the other extreme, the case when the set of all strong identities of a strong variety of partial algebras is invariant only under the identical replacement of operation symbols by terms. This leads to the concepts of unsolid and fluid varieties and some generalizations.