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Commuting polynomial operations of distributive lattices ; Couceiro, Miguel ; et al in Order: A Journal on the Theory of Ordered Sets and its Applications (2012), 29(2), 245-269 We describe which pairs of distributive lattice polynomial operations commute. Detailed reference viewed: 114 (1 UL)Partial orders induced by quasilinear clones Lehtonen, Erkko ; in Czermak, J.; Dorfer, G.; Eigenthaler, G. (Eds.) et al Proceedings of the Salzburg Conference 2011 (AAA81) (2012) We find sufficient conditions for a subclone C of Burle's clone and for a subclone of the polynomial clone of a finite semimodule to have the property that the associated C-minor partial order has finite ... [more ▼] We find sufficient conditions for a subclone C of Burle's clone and for a subclone of the polynomial clone of a finite semimodule to have the property that the associated C-minor partial order has finite principal ideals. We also prove that for these clones the C-minor partial order is universal for the class of countable partial orders whose principal ideals are finite. [less ▲] Detailed reference viewed: 57 (5 UL)Clones with finitely many relative R-classes Lehtonen, Erkko ; in Algebra Universalis (2011), 65(2), 109-159 For each clone C on a set A there is an associated equivalence relation analogous to Green's R-relation, which relates two operations on A if and only if each one is a substitution instance of the other ... [more ▼] For each clone C on a set A there is an associated equivalence relation analogous to Green's R-relation, which relates two operations on A if and only if each one is a substitution instance of the other using operations from C. We study the clones for which there are only finitely many relative R-classes. [less ▲] Detailed reference viewed: 37 (1 UL)The submaximal clones on the three-element set with finitely many relative R-classes Lehtonen, Erkko ; in Discussiones Mathematicae. General Algebra and Applications (2010), 30(1), 7-33 For each clone C on a set A there is an associated equivalence relation analogous to Green's R-relation, which relates two operations on A if and only if each one is a substitution instance of the other ... [more ▼] For each clone C on a set A there is an associated equivalence relation analogous to Green's R-relation, which relates two operations on A if and only if each one is a substitution instance of the other using operations from C. We study the maximal and submaximal clones on a three-element set and determine which of them have only finitely many relative R-classes. [less ▲] Detailed reference viewed: 95 (1 UL)Equivalence of operations with respect to discriminator clones Lehtonen, Erkko ; in Discrete Mathematics (2009), 309(4), 673-685 For each clone C on a set A there is an associated equivalence relation, called C-equivalence, on the set of all operations on A, which relates two operations iff each one is a substitution instance of ... [more ▼] For each clone C on a set A there is an associated equivalence relation, called C-equivalence, on the set of all operations on A, which relates two operations iff each one is a substitution instance of the other using operations from C. In this paper we prove that if C is a discriminator clone on a finite set, then there are only finitely many C-equivalence classes. Moreover, we show that the smallest discriminator clone is minimal with respect to this finiteness property. For discriminator clones of Boolean functions we explicitly describe the associated equivalence relations. [less ▲] Detailed reference viewed: 78 (2 UL) |
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