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Varieties of commutative residuated integral pomonoids and their residuation subreducts. (English) Zbl 0872.06007
Let $$\langle A;\oplus ,0,\leq \rangle$$ be a commutative (dually) integral partially ordered monoid whose identity $$0$$ is the least element of $$\langle A,\leq \rangle$$, where $$\leq$$ is a partial order compatible with the monoid operation $$\oplus$$ in the sense that $$a\oplus b\leq c\oplus d$$ whenever $$a\leq c$$ and $$b\leq d$$. If for each $$a,b\in A$$ there is a least element $$c\in A$$ (denoted $$a\div b$$) such that $$a\leq c\oplus b$$, then the resulting structure $$\langle A;\oplus ,\div ,0,\leq \rangle$$ is called a partially ordered commutative (dually) residuated (dually) integral monoid (briefly a pocrim). The operation $$\div$$ is called residuation. BCK-algebras are the residuation subreducts of pocrims. The simplest example of a pocrim is the set of ideals of a commutative ring with $$1$$ with respect to the ideal multiplication as the monoid operation and the (lattice) order of reversed set inclusion. Residuation is defined by $$I\div J=I:J$$.
The class of all pocrims is a quasivariety $$\mathcal{V}$$ which is not a variety, but it is relatively congruence distributive, has the relative congruence extension property, and is relatively point regular with respect to the monoid identity $$0$$. All $$0$$-regular subvarieties are congruence $$n$$-permutable for some $$n$$. In several varieties of pocrims (for example in the variety of Brouwerian semilattices and the variety of hoops) the finitely generated subdirectly irreducible algebras can be obtained from a finite number of simple algebras in the variety by a finite number of applications of the operations of variety generation and ordinal sum. Cancellative pocrims form a subquasivariety $$\mathcal{C}$$ which has a Mal’cev term and generates a congruence permutable variety (not contained in $$\mathcal{V}$$). The subvarieties of $$\mathcal{C}$$ form a lattice which has no greatest element.

##### MSC:
 06B20 Varieties of lattices 13A15 Ideals and multiplicative ideal theory in commutative rings 06F05 Ordered semigroups and monoids 06F35 BCK-algebras, BCI-algebras (aspects of ordered structures)
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##### References:
  K. Amer, 1982, Commutative monoids with monus, University of Waterloo · Zbl 0549.20039  Amer, K., Equationally complete classes of commutative monoids with monus, Algebra universalis, 18, 129-131, (1984) · Zbl 0549.20039  Birkhoff, G., Lattice theory, Colloquium publications, 25, (1984), Am. Math. Soc Providence  Blok, W.J.; Ferreirim, I.M.A., Hoops and their implicational reducts (abstract), Algebraic methods in logic and in computer science, (1993), Polish Academy of SciencesBanach Center PublicationsInstitute of Mathematics Warsaw · Zbl 0848.06013  W. J. Blok, I. M. A. Ferreirim, 1996, On the variety of hoops · Zbl 1012.06016  Blok, W.J.; Köhler, P.; Pigozzi, D., On the structure of varieties with equationally definable principal congruences II, Algebra universalis, 18, 334-379, (1984) · Zbl 0558.08001  Blok, W.J.; Pigozzi, D., On the structure of varieties with equationally definable principal congruences I, Algebra universalis, 15, 195-227, (1982) · Zbl 0512.08002  Blok, W.J.; Pigozzi, D., Local deduction theorems in algebraic logic, Algebraic logic, (1988), Colloquia Mathematica Societatis Janos Bolyai Budapest, p. 75-109 · Zbl 0751.03036  Blok, W.J.; Pigozzi, D., Algebraizable logics, Memoirs of the American mathematical society, 77, No. 396, (1989), Am. Math. Soc Providence · Zbl 0664.03042  Blok, W.J.; Pigozzi, D., On the structure of varieties with equationally definable principal congruences III, Algebra universalis, 32, 545-608, (1994) · Zbl 0817.08004  Blok, W.J.; Raftery, J.G., Failure of the congruence extension property inBCK, Math. japonica, 38, 633-638, (1993) · Zbl 0785.08006  Blok, W.J.; Raftery, J.G., On the quasivariety ofBCK, Algebra universalis, 33, 68-90, (1995) · Zbl 0818.06015  Bosbach, B., Komplementäre halbgruppen. axiomatik und arithmetik, Fund. math., 64, 257-287, (1969) · Zbl 0183.30603  Burris, S.; Sankappanavar, H.P., A course in universal algebra, Graduate texts in mathematics, 78, (1981), Springer-Verlag New York · Zbl 0478.08001  Chajda, I., Distributivity and modularity of lattices of tolerance relations, Algebra universalis, 12, 247-255, (1981) · Zbl 0469.08003  Chajda, I., Tolerances in congruence permutable algebras, Czechoslovak math. J., 38, 218-225, (1988) · Zbl 0678.08004  Chang, C.C., Algebraic analysis of many valued logics, Trans. amer. math. soc., 88, 467-490, (1958) · Zbl 0084.00704  Cornish, W.H., A large variety ofBCK, Math. japonica, 26, 339-344, (1981) · Zbl 0463.03039  Cornish, W.H., On Iséki’sBCK, Lecture notes in pure and applied mathematics, 74, (1982), Dekker New York, p. 101-122 · Zbl 0486.03033  Cornish, W.H., BCK, Math. japonica, 27, 63-73, (1982) · Zbl 0496.03045  Cornish, W.H., BCK, Math. japonica, 29, 439-447, (1984) · Zbl 0554.03034  Diego, A., Sur LES algèbres de Hilbert, Collection de logique mathematique, ser. A, No. 21, (1966), Gauthier-Villars Paris  Dyrda, K., None of the varietyE_n,n, Demonstratio math., 20, 215-219, (1987) · Zbl 0655.06013  Evans, T., Some connections between residual finiteness, finite embeddability and the word problem, J. London math. soc., 1, 399-403, (1969) · Zbl 0184.03502  I. M. A. Ferreirim, 1992, On Varieties and Quasivarieties of Hoops and Their Reducts, University of Illinois at Chicago  Fleischer, I., EveryBCK, J. algebra, 119, 360-365, (1988) · Zbl 0658.06012  Grätzer, G., Universal algebra, (1979), Springer-Verlag New York  J. Hagemann, June 1973, On Regular and Weakly Regular Congruences, Technische Hochschule Darmstadt  Hagemann, J.; Herrmann, C., A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity, Arch. math. (basel), 32, 234-245, (1979) · Zbl 0419.08001  Higgs, D., Dually residuated commutative monoids with identity element as least element do not form an equational class, Math. japonica, 29, 69-75, (1984) · Zbl 0549.06009  Idziak, P.M., On varieties ofBCK, Math. japonica, 28, 157-162, (1983) · Zbl 0518.06015  Iséki, K., An algebra related with a propositional calculus, Proc. Japan acad., 42, 26-29, (1966) · Zbl 0207.29304  Iséki, K., BCK, Math. japonica, 24, 107-119, (1979) · Zbl 0438.03059  Iséki, K., OnBCK, Math. japonica, 24, 625-626, (1980) · Zbl 0428.03055  Iséki, K.; Tanaka, S., An introduction to the theory ofBCK, Math. japonica, 23, 1-26, (1978) · Zbl 0385.03051  Kalman, J.A., Equational completeness and families of sets closed under subtraction, Nederl. acad. wetensch. proc. ser. A, 63, 402-406, (1960) · Zbl 0115.01301  Kearnes, K., Congruence permutable and congruence 3-permutable locally finite varieties, J. algebra, 156, 36-49, (1993) · Zbl 0790.08007  Köhler, P., Brouwerian semilattices, Trans. amer. math. soc., 268, 103-126, (1981) · Zbl 0473.06003  Krull, W., Axiomatische begruendung der allgemeinen ideal theorie, Sitzung der physikalisch-medicinische societaet zu erlangen, 56, 47-63, (1924)  McKenzie, R.N.; McNulty, G.F.; Taylor, W.F., Algebras, lattices, varieties, (1987), Wadsworth & Brooks/Cole Monterey · Zbl 0611.08001  Mitschke, A., Implication algebras are 3-permutable and 3-distributive, Algebra universalis, 1, 182-186, (1971) · Zbl 0242.08005  Monteiro, A., Cours sur LES algèbres de Hilbert et de Tarski, (1960), Instituto Mat. Univ. del Sur Bahia Blanca  Mundici, D., MVBCK, Math. japonica, 31, 889-894, (1986)  Ono, H.; Komori, Y., Logics without the contraction rule, J. symbolic logic, 50, 169-201, (1985) · Zbl 0583.03018  T. M. Owens, 1974, Varieties of Skolem Rings, Purdue University  Pałasinski, M., Some remarks onBCK, Math. seminar notes Kobe univ., 8, 137-144, (1980) · Zbl 0435.03048  Pałasinski, M., On ideal and congruence lattices ofBCK, Math. japonica, 26, 543-544, (1981) · Zbl 0476.03064  Pałasinski, M., An embedding theorem forBCK, Math. seminar notes Kobe univ., 10, 749-751, (1982)  M. Pałasiński, Subdirectly irreducibles inE_n-varieties ofBCK  M. Pałasiński, BCK  Pondělı́ček, B., On a certain class ofBCK, Math. japonica, 31, 775-782, (1986) · Zbl 0616.03044  Raftery, J.G.; Rosenberg, I.G.; Sturm, T., Tolerance relations andBCK, Math. japonica, 36, 399-410, (1991) · Zbl 0751.06012  Raftery, J.G.; Sturm, T., Tolerance numbers, congruencenBCK, Czechoslovak math. J., 42, 727-740, (1992) · Zbl 0784.08002  Sturm, T., On commutativeBCKBCK, Math. japonica, 27, 197-212, (1982)  T. Sturm  Tanaka, S., On ∧-commutative algebras, Math. seminar notes Kobe univ., 3, 59-64, (1975)  Traczyk, T., On the variety of bounded commutativeBCK, Math. japonica, 24, 283-292, (1979) · Zbl 0422.03038  Traczyk, T., On the structure ofBCKzxyxzyxy, Math. japonica, 33, 319-324, (1988)  Ward, M.; Dilworth, R.P., Residuated lattices, Trans. amer. math. soc., 45, 335-354, (1939) · Zbl 0021.10801  Wroński, A., BCK, Math. japonica, 28, 211-213, (1983) · Zbl 0518.06014  Wroński, A., An algebraic motivation forBCK, Math. japonica, 30, 187-193, (1985) · Zbl 0569.03029  Wroński, A.; Kabziński, J.K., There is no largest variety ofBCK, Math. japonica, 29, 545-549, (1984) · Zbl 0542.03043
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