Unsolid and fluid strong varieties of partial algebras

S. Busaman and K. Denecke

International Journal of Mathematics and Mathematical Sciences, 2006, vol. 2006, 1-19

Abstract:

A partial algebra 𝒜 = ( A ; ( f i A ) i ∈ I ) consists of a set A and an indexed set ( f i A ) i ∈ I of partial operations f i A : A n i ⊸ → A . Partial operations occur in the algebraic description of partial recursive functions and Turing machines. A pair of terms p ≈ q over the partial algebra 𝒜 is said to be a strong identity in 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 ≈ 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.

Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:068534

DOI: 10.1155/IJMMS/2006/68534

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