Abstract
Aiming to understand equivalence relations that model concurrent computation, we investigate congruences of multinomial lattices \(\mathcal{L}(v)\) introduced by Bennett and Birkhoff (Algebra Univers. 32(1):115–144, 1994). Our investigation gives rise to an explicit description of the join dependency relation between two join irreducible elements and of its reflexive transitive closure. The explicit description emphasizes several properties and makes it possible to separate the equational theories of multinomial lattices by their dimensions. In their covering of non modular varieties Jipsen and Rose (Varieties of lattices, Lecture Notes in Mathematics, vol. 1533, Springer, Berlin, 1992) define a sequence of equations \(SD_{n}(\land)\), for n ≥ 0. Our main result sounds as follows: if \(v = (v_{1},\ldots,v_{n}) \in \mathbb{N}^{n}\) and v i > 0 for i = 1,...,n, then the multinomial lattice \(\mathcal{L}(v)\) satisfies \(SD_{n-1}(\land)\) and fails \(SD_{n-2}(\land)\)
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Arnold, A.: An extension of the notions of traces and of asynchronous automata. ITA 25, 355–396 (1991)
Barbut, M., Monjardet, B.: Ordre et Classification: Algèbre et Combinatoire. Tomes I et II. Librairie Hachette, Paris, 1970. Méthodes Mathématiques des Sciences de l’Homme, Collection Hachette Université (1970)
Barr, M.: The separated extensional Chu category. Theory Appl. Categ. 4(6), 137–147 (1998) (electronic)
Bennett, M.K., Birkhoff, G.: Two families of Newman lattices. Algebra Univers. 32(1), 115–144 (1994)
Birkhoff, G.: Lattice theory. American Mathematical Society Colloquium Publications, vol. 25, 3rd edn. American Mathematical Society, Providence, RI (1979)
Björner, A.: Orderings of Coxeter groups. In: Combinatorics and algebra (Boulder, Colo., 1983). of Contemp. Math., vol. 34, pp. 175–195. Amer. Math. Soc., Providence, RI (1984)
Caspard, N.: The lattice of permutations is bounded. Internat. J. Algebra Comput. 10(4), 481–489 (2000)
Caspard, N., Le Conte de Poly-Barbut, C., Morvan, M.: Cayley lattices of finite Coxeter groups are bounded. Adv. in Appl. Math. 33(1), 71–94 (2004)
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, New York (2002)
Day, A.: Characterizations of finite lattices that are bounded-homomorphic images of sublattices of free lattices. Canad. J. Math. 31(1), 69–78 (1979)
Diekert, V., Rozenberg, G.: In: The Book of Traces. World Scientific Publishing Co. Inc., River Edge, NJ (1995)
Duquenne, V., Cherfouh, A.: On permutation lattices. Math. Social Sci. 27(1), 73–89 (1994)
Ferrari, L., Pinzani, R.: Lattices of lattice paths. J. Stat. Plan. Inference 135(1), 77–92 (2005)
Flath, S.: The order dimension of multinomial lattices. Order 10(3), 201–219 (1993)
Freese, R., Ježek, J., Nation, J.B.: Free lattices. Mathematical Surveys and Monographs, vol. 42. American Mathematical Society, Providence, RI (1995)
Ganter, B., Wille, R.: Formal Concept Analysis. Springer, Berlin (1999), Mathematical foundations, Translated from the 1996 German original by Cornelia Franzke
Goubault, É, Raussen, M.: Dihomotopy as a tool in state space analysis. LATIN 2002: Theoretical Informatics (Cancun). Lecture Notes in Comput. Sci., vol. 2286, pp. 16–37. Springer, Berlin (2002)
Grätzer, G.: General Lattice Theory, 2nd Edn. Birkhäuser Verlag, Basel (1998)
Guilbaud, G., Rosenstiehl, P.: Analyse algébrique d’un scrutin. Math. Sci. Hum. (4), 9–33 (1963)
Hoogers, P.W., Kleijn, H.C.M., Thiagarajan, P.S.: An event structure semantics for general Petri nets. Theoret. Comput. Sci. 153(1–2), 129–170 (1996)
Jipsen, P., Rose, H.: Varieties of lattices. Lecture Notes in Mathematics, vol. 1533. Springer, Berlin (1992)
Jónsson, B., Nation, J.B.: A report on sublattices of a free lattice. In: Contributions to universal algebra (Colloq., József Attila Univ., Szeged, 1975). Colloq. Math. Soc. János Bolyai, vol. 17, pp. 223–257. North-Holland, Amsterdam (1977)
Krattenthaler, C.: The enumeration of lattice paths with respect to their number of turns. In: Advances in combinatorial methods and applications to probability and statistics, pp. 29–58. Birkhäuser, Boston, MA (1997)
Le Conte de Poly-Barbut, C.: Automorphismes du permutoèdre et votes de Condorcet. Math. Inf. Sci. Hum. 111, 73–82 (1990)
Le Conte de Poly-Barbut, C.: Le diagramme du treillis permutoèdre est intersection des diagrammes de deux produits directs d’ordres totaux. Math. Inf. Sci. Hum. 112, 49–53 (1990)
Markowsky, G.: Permutation lattices revisited. Math. Social Sci. 27(1), 59–72 (1994)
Mohanty, S.G.: Lattice Path Counting and Applications. Academic, New York (1979). Probability and Mathematical Statistics
Nation, J.B.: An approach to lattice varieties of finite height. Algebra Univers. 27(4), 521–543 (1990)
Reading, N.: Lattice congruences of the weak order. Order 21(4), 315–344 (2004)
Semënova, M.V.: On lattices that are embeddable into lattices of suborders. Algebra Log. 44(4), 483–511, 514 (2005)
Wehrung, F.: The dimension monoid of a lattice. Algebra Univers. 40(3), 247–411 (1998)
Wehrung, F.: From join-irreducibles to dimension theory for lattices with chain conditions. J. Algebra Appl. 1(2), 215–242 (2002)
Yanagimoto, T., Okamoto, M.: Partial orderings of permutations and monotonicity of a rank correlation statistic. Ann. Inst. Statist. Math. 21, 489–506 (1969)
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Santocanale, L. On the Join Dependency Relation in Multinomial Lattices. Order 24, 155–179 (2007). https://doi.org/10.1007/s11083-007-9066-0
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DOI: https://doi.org/10.1007/s11083-007-9066-0