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**On the structure of varieties with equationally definable principal congruences. IV.**
*(English)*
Zbl 0817.08005

This is the fourth part of a series of papers dealing with the structure of varieties with equationally definable principal congruences (EDPC) and the connection with algebraic logic [see also the preceding review of Part III].

The fixed-point ternary discriminator function, the ternary (commutative and regular) deductive terms (TD) were investigated in the third paper. In this paper, the authors investigate a certain assertional logic that is inherent in every variety with a commutative, regular TD term. The notion of (pseudo-interior operation and) pseudo-interior algebra is introduced in Definitions 2.1 and 2.6, and the elementary parts of its arithmetic are developed. The authors also investigate the notion of open filter (Definition 2.13), and Theorem 2.16 establishes the fundamental relationship between open filters and congruences on a pseudo-interior algebra with compatible operations. Theorem 3.1 gives an equational characterization of pseudo-interior algebras. The main result of the paper is Theorem 4.1 stating that (roughly speaking) the class of compact homomorphic images of an algebra has a commutative, regular TD term iff the algebra itself is termwise definitionally equivalent to a pseudo- interior algebra with compatible operations. The last section contains some open problems.

The fixed-point ternary discriminator function, the ternary (commutative and regular) deductive terms (TD) were investigated in the third paper. In this paper, the authors investigate a certain assertional logic that is inherent in every variety with a commutative, regular TD term. The notion of (pseudo-interior operation and) pseudo-interior algebra is introduced in Definitions 2.1 and 2.6, and the elementary parts of its arithmetic are developed. The authors also investigate the notion of open filter (Definition 2.13), and Theorem 2.16 establishes the fundamental relationship between open filters and congruences on a pseudo-interior algebra with compatible operations. Theorem 3.1 gives an equational characterization of pseudo-interior algebras. The main result of the paper is Theorem 4.1 stating that (roughly speaking) the class of compact homomorphic images of an algebra has a commutative, regular TD term iff the algebra itself is termwise definitionally equivalent to a pseudo- interior algebra with compatible operations. The last section contains some open problems.

Reviewer: E.Fried (Budapest)

### Keywords:

ternary deductive term; structure of varieties; equationally definable principal congruences; algebraic logic; assertional logic; pseudo- interior algebra; open filter; congruences; compact homomorphic images
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\textit{W. J. Blok} and \textit{D. Pigozzi}, Algebra Univers. 31, No. 1, 1--35 (1994; Zbl 0817.08005)

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### References:

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