Department of Mathematics
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In 1954, Mal?cev proved a remarkable characterization of varieties V such that the composition of two congruences is always their join (equivalently, any two congruences permute). The theorem says that the equational theory of V contains a ternary operation m(x, y, z) such that m(x, x, y) = y and m(x, y, y) = x for each x, y. In 1963, Alden Pixley gave a similar characterization for varieties having both distributive and permutable congruences. Later on, further similar Mal'cev-type conditions were found. For instance, Bjarni Jonsson characterized congruence distributive varieties in 1967 and A. Day described congruence modular ones in 1969. Groups, rings, lattices and Boolean algebras are motivating examples of such varieties. In this talk, we show how some of these Mal'cev terms can be successfully employed to construct minimal equational bases satisfying some additional syntactic properties. This ongoing research has been done in collaboration with David Kelly, Bill McCune, Bob Quackenbush and Barry Wolk. The phrase "Malcev-type condition" was first used by George Grätzer in his characterization of regular varieties.

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2. Padmanabhan, R.; Quackenbush, R. W. Equational theories of algebras with distributive congruences. Proc. Amer. Math. Soc. 41 (1973), 373–377.
3. Padmanabhan, R. Equational theory of algebras with a majority polynomial. Algebra Universalis 7 (1977), no. 2, 273–275.
4. Padmanabhan, R.; Wolk, B. Equational theories with a minority polynomial. Proc. Amer. Math. Soc. 83 (1981), no. 2, 238–242.
5. Padmanabhan, R.; McCune, W. Single identities for ternary Boolean algebras. Comput. Math. Appl. 29 (1995), no. 2, 13–16.
6. Kelly, David; Padmanabhan, R. Irredundant self-dual bases for self-dual lattice varieties. Algebra Universalis 52 (2005), no. 4, 501–517.