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**1 - 2**of**2**### CATEGORICAL EQUIVALENCE OF ALGEBRAS WITH A MAJORITY TERM

, 1998

"... Abstract. Let A be a finite algebra with a majority term. We characterize those algebras categorically equivalent to A. The description is in terms of a derived structure with universe consisting of all subalgebras of A × A, and with operations of composition, converse and intersection. The main the ..."

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Abstract. Let A be a finite algebra with a majority term. We characterize those algebras categorically equivalent to A. The description is in terms of a derived structure with universe consisting of all subalgebras of A × A, and with operations of composition, converse and intersection. The main theorem is used to get a different sort of characterization of categorical equivalence for algebras generating an arithmetical variety. We also consider clones of co-height at most two. In addition, we provide new proofs of several characterizations in the literature, including quasi-primal, lattice-primal and congruence-primal algebras. Majority operations have long held a special place in universal algebra. It has been known for quite some time that any variety of algebras possessing a majority term is congruence distributive. In 1975, Baker and Pixley discovered that for a finite algebra A with a majority term, the set of subalgebras of A2 completely determines the term operations on A. In addition, every subalgebra of Ak (with k ≥ 2) is completely determined by all of its 2-fold projections. Conversely, G. Bergman

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"... Remarks on categorical equivalence of finite unary algebras 1. Background M. Krasner’s original theorems from 1939 say that a finite algebra A (1) is an essentially multiunary algebra in which all operations are permutations if and only if for every n, Sub(A n) is closed under (settheoretic) complem ..."

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Remarks on categorical equivalence of finite unary algebras 1. Background M. Krasner’s original theorems from 1939 say that a finite algebra A (1) is an essentially multiunary algebra in which all operations are permutations if and only if for every n, Sub(A n) is closed under (settheoretic) complementation; (2) is an essentially multiunary algebra if and only if for every n, Sub(A n) is closed under unions. By an “essentially multiunary algebra ” I mean an algebra in which each basic operation is essentially unary, (i.e., depends on only one variable). From now on, I’ll just refer to these as unary algebras. The algebras in the first part of Krasner’s theorem will be called “group actions”. Motivated by number (1), I proved the following (see attached manuscript “Boolean Krasner Algebras”).