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Projective algebras and primitive subquasivarieties in varieties with factor congruences

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Abstract

We prove that in the varieties where every compact congruence is a factor congruence and every nontrivial algebra contains a minimal subalgebra, a finitely presented algebra is projective if and only if it has every minimal algebra as its homomorphic image. Using this criterion of projectivity, we describe the primitive subquasivarieties of discriminator varieties that have a finite minimal algebra embedded in every nontrivial algebra from this variety. In particular, we describe the primitive quasivarieties of discriminator varieties of monadic Heyting algebras, Heyting algebras with regular involution, Heyting algebras with a dual pseudocomplement, and double-Heyting algebras.

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Acknowledgements

First of all, the author is grateful to V. Marra, whose suggestion prompted the extension of the results from \(\varvec{\mathsf {WS}5}\)-algebras to discriminator varieties. The author also wishes to thank K. Adaricheva, W. Dzik, V. Meskhi, A. Nurakunov and especially H.P. Sankappanavar for the fruitful discussions. Additionally, the author is indebted to the referees, whose suggestions greatly facilitated the process of improving the paper.

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  1. Metropolitan Telecommunications, 55 Water St., 32 Fl., New York, NY, 10041, USA

    Alex Citkin

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  1. Alex Citkin
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Correspondence to Alex Citkin.

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Presented by K. Kearnes.

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Citkin, A. Projective algebras and primitive subquasivarieties in varieties with factor congruences. Algebra Univers. 79, 66 (2018). https://doi.org/10.1007/s00012-018-0555-3

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