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See detailConservative median algebras and semilattices
Couceiro, Miguel; Marichal, Jean-Luc UL; Teheux, Bruno UL

in Order : A Journal on the Theory of Ordered Sets and its Applications (2016), 33(1), 121-132

We characterize conservative median algebras and semilattices by means of forbidden substructures and by providing their representation as chains. Moreover, using a dual equivalence between median ... [more ▼]

We characterize conservative median algebras and semilattices by means of forbidden substructures and by providing their representation as chains. Moreover, using a dual equivalence between median algebras and certain topological structures, we obtain descriptions of the median-preserving mappings between products of finitely many chains. [less ▲]

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See detailParametrized arity gap
Couceiro, Miguel UL; Lehtonen, Erkko UL; Waldhauser, Tamás UL

in Order : A Journal on the Theory of Ordered Sets and its Applications (2013), 30(2), 557-572

We propose a parametrized version of arity gap. The parametrized arity gap gap(f,l) of a function f: Aⁿ → B measures the minimum decrease in the number of essential variables of f when l consecutive ... [more ▼]

We propose a parametrized version of arity gap. The parametrized arity gap gap(f,l) of a function f: Aⁿ → B measures the minimum decrease in the number of essential variables of f when l consecutive identifications of pairs of essential variables are performed. We determine gap(f,l) for an arbitrary function f and a positive integer l. We also propose other variants of arity gap and discuss further problems pertaining to the effect of identification of variables on the number of essential variables of functions. [less ▲]

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See detailOn the Poset of Computation Rules for Nonassociative Calculus
Couceiro, Miguel UL; Grabisch, Michel

in Order : A Journal on the Theory of Ordered Sets and its Applications (2012), 30(1), 269-288

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See detailCommuting polynomial operations of distributive lattices
Behrisch, Mike; Couceiro, Miguel UL; Kearnes, Keith A. 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.

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See detailAssociative polynomial functions over bounded distributive lattices
Couceiro, Miguel UL; Marichal, Jean-Luc UL

in Order : A Journal on the Theory of Ordered Sets and its Applications (2011), 28(1), 1-8

The associativity property, usually defined for binary functions, can be generalized to functions of a given fixed arity n>=1 as well as to functions of multiple arities. In this paper, we investigate ... [more ▼]

The associativity property, usually defined for binary functions, can be generalized to functions of a given fixed arity n>=1 as well as to functions of multiple arities. In this paper, we investigate these two generalizations in the case of polynomial functions over bounded distributive lattices and present explicit descriptions of the corresponding associative functions. We also show that, in this case, both generalizations of associativity are essentially the same. [less ▲]

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See detailOn the homomorphism order of labeled posets
Kwuida, Léonard; Lehtonen, Erkko UL

in Order : A Journal on the Theory of Ordered Sets and its Applications (2011), 28(2), 251-265

Partially ordered sets labeled with k labels (k-posets) and their homomorphisms are examined. We give a representation of directed graphs by k-posets; this provides a new proof of the universality of the ... [more ▼]

Partially ordered sets labeled with k labels (k-posets) and their homomorphisms are examined. We give a representation of directed graphs by k-posets; this provides a new proof of the universality of the homomorphism order of k-posets. This universal order is a distributive lattice. We investigate some other properties, namely the infinite distributivity, the computation of infinite suprema and infima, and the complexity of certain decision problems involving the homomorphism order of k-posets. Sublattices are also examined. [less ▲]

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See detailDescending chains and antichains of the unary, linear, and monotone subfunction relations
Lehtonen, Erkko UL

in Order : A Journal on the Theory of Ordered Sets and its Applications (2006), 23(2-3), 129-142

The C-subfunction relations on the set of operations on a finite base set A defined by function classes C are examined. For certain clones C on A, it is determined whether the partial orders induced by ... [more ▼]

The C-subfunction relations on the set of operations on a finite base set A defined by function classes C are examined. For certain clones C on A, it is determined whether the partial orders induced by the respective C-subfunction relations have infinite descending chains or infinite antichains. More specifically, we investigate the subfunction relations defined by the clone of all functions on A, the clones of essentially at most unary operations, the clones of linear functions on a finite field, and the clones of monotone functions with respect to the various partial orders on A. [less ▲]

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See detailAggregation on finite ordinal scales by scale independent functions
Marichal, Jean-Luc UL; Mesiar, Radko

in Order : A Journal on the Theory of Ordered Sets and its Applications (2004), 21(2), 155-180

We define and investigate the scale independent aggregation functions that are meaningful to aggregate finite ordinal numerical scales. Here scale independence means that the functions always have ... [more ▼]

We define and investigate the scale independent aggregation functions that are meaningful to aggregate finite ordinal numerical scales. Here scale independence means that the functions always have discrete representatives when the ordinal scales are considered as totally ordered finite sets. We also show that those scale independent functions identify with the so-called order invariant functions, which have been described recently. In particular, this identification allows us to justify the continuity property for certain order invariant functions in a natural way. [less ▲]

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