×

An easy test for congruence modularity. (English) Zbl 1259.08004

Let \(\Sigma\) be a set of identities. \(\Sigma\) is inconsistent if \(\Sigma \models x \approx y\), otherwise \(\Sigma\) is consistent. \(\Sigma\) is idempotent if for every function symbol \(F\) appearing in \(\Sigma\), it is the case that \(\Sigma \models F(x, x,..., x) \approx x\). A term \(t\) is linear if it has at most one occurrence of a function symbol. An identity \(s \approx t\) is linear if both \(s\) and \(t\) are linear.
There are two well-known theorems, A. Day’s theorem [Can. Math. Bull. 12, 167–173 (1969; Zbl 0181.02302)] and H. P. Gumm’s theorem [Arch. Math. 36, 569–576 (1981; Zbl 0465.08005)], which each characterize congruence modularity. The authors of the paper under review describe an easy way to determine whether the realization of a set of idempotent identities guarantees congruence modularity or the satisfaction of a non-trivial congruence identity. Namely, for a set \(\Sigma\) of idempotent identities, they define the notion of a derivative, \(\Sigma '\), which is a superset of idempotent identities in the same language. One of their main results is that \(\Sigma\) axiomatizes a congruence-modular variety if the derivative of \(\Sigma\) is inconsistent. Another main result is that \(\Sigma\) axiomatizes a variety that satisfies some nontrivial congruence identity if its \(n\)-th derivative is inconsistent for some \(n\). The two final results show that for a set \(\Sigma\) of idempotent linear identities, the derivative test is a necessary and sufficient condition to determine if \(\Sigma\) defines a variety that is congruence-modular or satisfies a nontrivial congruence identity.

MSC:

08B10 Congruence modularity, congruence distributivity
08B05 Equational logic, Mal’tsev conditions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bentz W.: Topological implications in varieties. Algebra Universalis 42, 9–16 (1999) · Zbl 0977.08005 · doi:10.1007/s000120050120
[2] Berman J., Idziak P., Marković P., McKenzie R., Valeriote M., Willard R.: Varieties with few subalgebras of powers. Trans. Amer. Math. Soc. no. 3, (362), 1445–1473 (2010) · Zbl 1190.08004 · doi:10.1090/S0002-9947-09-04874-0
[3] Czédli G., Freese R.: On congruence distributivity and modularity. Algebra Universalis 17(no.2), 216–219 (1983) · Zbl 0548.08003 · doi:10.1007/BF01194531
[4] Day A.: A characterization of modularity for congruence lattices of algebras. Canad. Math. Bull. 12, 167–173 (1969) · Zbl 0181.02302 · doi:10.4153/CMB-1969-016-6
[5] Day A.: p-modularity implies modularity in equational classes. Algebra Universalis 3, 398–399 (1973) · Zbl 0288.06012 · doi:10.1007/BF02945142
[6] Freese R., Nation J.B.: Congruence lattices of semilattices. Pacific J. Math. 49, 51–58 (1973) · Zbl 0287.06002 · doi:10.2140/pjm.1973.49.51
[7] Gedeonová E.: A characterization of p-modularity for congruence lattices of algebras. Acta Fac. Rerum Natur. Univ. Comenian. Math. Publ. 28, 99–106 (1972) · Zbl 0264.06008
[8] Gumm H.-P.: Congruence modularity is permutability composed with distributivity. Arch. Math. (Basel) 36, 569–576 (1981) · Zbl 0465.08005 · doi:10.1007/BF01223741
[9] Hagemann J., Mitschke A.: On n-permutable congruences. Algebra Universalis 3, 8–12 (1973) · Zbl 0273.08001 · doi:10.1007/BF02945100
[10] Hobby, D., McKenzie, R.: The Structure of Finite Algebras. Contemporary Mathematics, vol. 76. American Mathematical Society, Providence (1988) · Zbl 0721.08001
[11] Jónsson B.: On the representation of lattices. Math. Scand. 1, 193–206 (1953) · Zbl 0053.21304
[12] Jónsson B.: Congruence varieties. Algebra Universalis 10, 355–394 (1980) · Zbl 0438.08003 · doi:10.1007/BF02482916
[13] Kearnes K.A.: Almost all minimal idempotent varieties are congruence modular. Algebra Universalis 44, 39–45 (2000) · Zbl 1014.08009 · doi:10.1007/s000120050169
[14] Kearnes K.A.: Congruence join semidistributivity is equivalent to a congruence identity. Algebra Universalis 46(no. 3), 373–387 (2001) · Zbl 1063.08008 · doi:10.1007/PL00000351
[15] Kearnes, K.A., Kiss, E.W.: The Shape of Congruence Lattices. Mem. Amer. Math. Soc., to appear · Zbl 1294.08002
[16] Kearnes, K.A., Kiss, E.W., Szendrei, Á.: Growth rates of finite algebras (2011, manuscript) · Zbl 1320.08002
[17] Kearnes K.A., Nation J.B.: Axiomatizable and nonaxiomatizable congruence prevarieties. Algebra Universalis 59, 323–335 (2008) · Zbl 1165.08002 · doi:10.1007/s00012-008-2068-y
[18] Kearnes K.A., Sequeira L.: Hausdorff properties of topological algebras. Algebra Universalis 47, 343–366 (2002) · Zbl 1064.08005 · doi:10.1007/s00012-002-8194-z
[19] Kelly, D.: Basic equations: word problems and Mal’cev conditions. Abstract 701-08-4, Notices Amer. Math. Soc. 20, A-54 (1973)
[20] Lipparini P.: Congruence identities satisfied in n-permutable varieties. Boll. Un. Mat. Ital. B (7) 8, (no. 4), 851–868 (1994) · Zbl 0816.08007
[21] Lipparini P.: n-permutable varieties satisfy nontrivial congruence identities. Algebra Universalis 33, 159–168 (1995) · Zbl 0821.08007 · doi:10.1007/BF01190927
[22] Lipparini, P.: Every m-permutable variety satisfies the congruence identity {\(\alpha\)}{\(\beta\)} h = {\(\alpha\)}{\(\gamma\)} h . Proc. Amer. Math. Soc. 136, no. 4, 1137–1144 (2008) · Zbl 1132.08001
[23] McNulty G.F.: Undecidable properties of finite sets of equations. J. Symbolic Logic 41, (no. 3), 589–604 (1976) · Zbl 0375.02040 · doi:10.2307/2272037
[24] Nation J.B.: Varieties whose congruences satisfy certain lattice identities. Algebra Universalis 4, 78–88 (1974) · Zbl 0299.08002 · doi:10.1007/BF02485709
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.