Abstract
AbstractMeet semidistributive varieties are in a sense the last of the most important classes in universal algebra for which it is unknown whether it can be characterized by a strong Maltsev condition. We present a new, relatively simple Maltsev condition characterizing the meet-semidistributive varieties, and provide a candidate for a strong Maltsev condition.
Highlights
The tame congruence theory (TCT) [4], a structure theory of general finite algebras, has revealed that there are only 5 possibly local behaviors of a finite algebra:(1) algebra having only unary functions, (2) one-dimensional vector space, (3) the two-element boolean algebra, (4) the two-element lattice, (5) the two element semilattice.If there is a local behavior of type (i) in an algebra A, the algebra is said to have type (i)
The set of “bad” types that are omitted in a variety is an important structural information; for instance, it plays a significant role in the fixed-template constraint satisfaction problem [3]
Types in the TCT are defined only for locally finite varieties, the type-omitting classes have alternative characterizations which do not refer to the type-set
Summary
The tame congruence theory (TCT) [4], a structure theory of general finite algebras, has revealed that there are only 5 possibly local behaviors of a finite algebra:. [9] A locally finite variety V omits type (1) if and only if there is an idempotent WNU (weak near unanimity) term in A, that is a term satisfying the following identities:. A locally finite variety omits types (1) and (2) if and only if it has three-ary and four-ary idempotent terms w3, w4 satisfying equations w3(yxx) = w3(xyx) = w3(xxy) = w4(yxxx). Types in the TCT are defined only for locally finite varieties (because only finite algebras are assigned types), the type-omitting classes have alternative characterizations which do not refer to the type-set. They are shown in the following table taken from [8]. Is there a strong Maltsev condition that is equivalent to congruence meet-semidistributivity?
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