# Article

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Keywords:
regular variety; subregular variety; deductive system; congruence class; difference system
Summary:
An algebra ${\mathcal A}= (A,F)$ is subregular alias regular with respect to a unary term function $g$ if for each $\Theta , \Phi \in \text{Con}\,{\mathcal A}$ we have $\Theta = \Phi$ whenever $[g(a)]_{\Theta } = [g(a)]_{\Phi }$ for each $a\in A$. We borrow the concept of a deductive system from logic to modify it for subregular algebras. Using it we show that a subset $C\subseteq A$ is a class of some congruence on $\Theta$ containing $g(a)$ if and only if $C$ is this generalized deductive system. This method is efficient (needs a finite number of steps).
References:
[1] Barbour G. D., Raftery J. G.: On the degrees of permutability of subregular varieties. Czechoslovak Math. J. 47 (1997), 317–325. DOI 10.1023/A:1022873713616 | MR 1452422
[2] Bělohlávek R., Chajda I.: Congruence classes in regular varieties. Acta Math. Univ. Comenian. (Bratislava) 68 (1999), 71–75. MR 1711075
[3] Bělohlávek R., Chajda I.: Relative deductive systems and congruence classes. Mult.- Valued Log. 5 (2000), 259–266. MR 1784274
[4] Blok W., Köhler P., Pigozzi D.: On the structure of varieties with equationally definable principal congruences II. Algebra Univers. 18 (1984), 334–379. MR 0745497
[5] Chajda I.: Congruence kernels in weakly regular varieties. Southeast Asian Bull. Math. 24 (2000), 15–18. DOI 10.1007/s10012-000-0015-8 | MR 1811209 | Zbl 0988.08002
[6] Chajda I., Rachůnek J.: Relational characterization of permutable and $n$-permutable varieties. Czechoslovak Math. J. 33 (1983), 505–508. MR 0721079
[7] Gumm H.-P., Ursini A.: Ideals in universal algebras. Algebra Univers. 19 (1984), 45–54. DOI 10.1007/BF01191491 | MR 0748908
[8] Ursini A.: Sulla varietá di algebre con una buona teoria degli ideali. Boll. Unione Mat. Ital. 6 (1972), 90–95. MR 0314728

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