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On the Join Dependency Relation in Multinomial Lattices

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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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Correspondence to Luigi Santocanale.

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Research supported by the project ANR SOAPDC.

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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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