ON COATOMS AND COMPLEMENTS IN CONGRUENCE LATTICES OF UNARS WITH MAL’TSEV OPERATION

One important problem is studying of lattices that naturally associated with universal algebra. In this article is considered algebras ⟨A, p, f⟩ with one Mal’tsev operation p and one unary operation f acting as endomorphism with respect to operation p. We study properties of congruence lattices of algebras ⟨A, p, f⟩ with Mal’tsev operation p that introduced by V. K. Kartashov. This algebra is defined as follows. Let ⟨A, f⟩ be an arbitrary unar and x, y ∈ A. For any element x of the unar ⟨A, f⟩ by f n (x) we denote the result of f applied n times to an element x. Also f 0 (x) = x. Assume that Mx,y = {n ∈ N ∪ {0} | f n (x) = f n (y)} and also k(x, y) = min Mx,y, if Mx,y ̸= ∅ and k(x, y) = ∞, if Mx,y = ∅. Assume further p(x, y, z) def = { z, if k(x, y) 6 k(y, z) x, if k(x, y) > k(y, z). It is described a structure of coatoms in congruence lattices of algebras ⟨A, p, f⟩ from this class. It is proved congruence lattices of algebras ⟨A, p, f⟩ has no coatoms if and only if the unar ⟨A, f⟩ is connected, contains one-element subunar and has infinite depth. In other cases congruence lattices of algebras ⟨A, p, f⟩ has uniquely coatom. It is showed for any congruences θ ̸= A × A and φ ̸= A × A of algebra ⟨A, p, f⟩ fulfills θ ∨ φ < A × A. Necessary and sufficient conditions when a congruence lattice of algebras from given class is complemented, uniquely complemented, relatively complemented, Boolean, generalized Boolean, geometric are obtained. It is showed any non-trivial congruence of algebra ⟨A, p, f⟩ from this class has no complement. It is proved that congruence lattices of any algebra ⟨A, p, f⟩ from given class is dual pseudocomplemented lattice.

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