### Unification in primal algebras, their powers and their varietiesUnification in primal algebras, their powers and their varieties

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 Author Nipkow, Tobias Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©1990 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Abstract This paper examines the unification problem in the class of primal algebras and the varieties they generate. An algebra is called primal if every function on its carrier can be expressed just in terms of the basic operations of the algebra. The two-element Boolean algebra is the simplest nontrivial example: Every truth-function can be realized in terms of the basic connectives, for example, negation and conjunction.It is shown that unification in primal algebras is unitary, that is, if an equation has a solution, it has a single most general one. Two unification algorithms, based on equation-solving techniques for Boolean algebras due to Boole and Lo¨wenheim, are studied in detail. Applications include certain finite Post algebras and matrix rings over finite fields. The former are algebraic models for many-valued logics, the latter cover in particular modular arithmetic.Then unification is extended from primal algebras to their direct powers, which leads to unitary unification algorithms covering finite Post algebras, finite, semisimple Artinian rings, and finite, semisimple nonabelian groups.Finally the fact that the variety generated by a primal algebra coincides with the class of its subdirect powers is used. This yields unitary unification algorithms for the equational theories of Post algebras and $\textit{p}-rings.$ ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 1990-10-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 37 Issue Number 4 Page Count 35 Starting Page 742 Ending Page 776

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Source: ACM Digital Library