Unsolid and fluid strong varieties of partial algebras

Abstract

A partial algebra𝒜=(A;(fiA)i∈I)consists of a setAand an indexed set(fiA)i∈Iof partial operationsfiA:Ani⊸→A. Partial operations occur in the algebraic description of partial recursive functions and Turing machines. A pair of termsp≈qover the partial algebra𝒜is said to be a strong identity in𝒜if the right-hand side is defined whenever the left-hand side is defined and vice versa, and both are equal. A strong identityp≈qis called a strong hyperidentity if when the operation symbols occurring inpandqare 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.