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April 21, 2022

Colloquium- Congruence permutability is prime- SC- 5211

A variety or equational class is a class of all algebras (algebraic structures of a given signature) satisfying a given set of identities. We say that a variety K interprets in a variety V if K has a set of identities such that if we replace the operation symbols of K with terms of V, then the obtained set of identities hold in V. As an example, the variety of groups interprets in the variety of rings (using its additive structure). As easily seen, interpretability is a quasiorder on the class of varieties. The blocks of this quasiorder are called the interpretability types. Garcıa and Taylor introduced the lattice of interpretability types of varieties that is obtained by taking the quotient of the class of varieties quasiordered by interpretability and the corresponding equivalence relation. We call a variety V congruence permutable if the variety K defined by m(x,y,y)=x and m(y,y,x)=x interprets in V. For example, the varieties of groups, rings and vector spaces are congruence permutable. In 1984 Garcia and Taylor formulated the conjecture that congruence permutability is a prime element in the lattice of interpretability types. We will give combinatorial proof of this conjecture and settle it in its full generality. This is joint work with G. Gyenize and L. Zadori.