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The Minor Order of Homomorphisms via Natural Dualities Poiger, Wolfgang ; Teheux, Bruno in Order: A Journal on the Theory of Ordered Sets and its Applications (2022) We study the minor relation for algebra homomorphims in finitely generated quasivarieties that admit a logarithmic natural duality. We characterize the minor homomorphism posets of finite algebras in ... [more ▼] We study the minor relation for algebra homomorphims in finitely generated quasivarieties that admit a logarithmic natural duality. We characterize the minor homomorphism posets of finite algebras in terms of disjoint unions of dual partition lattices and investigate reconstruction problems for homomorphisms. [less ▲] Detailed reference viewed: 35 (5 UL)Associative, idempotent, symmetric, and order-preserving operations on chains Devillet, Jimmy ; Teheux, Bruno in Order: A Journal on the Theory of Ordered Sets and its Applications (2020), 37(1), 45-58 We characterize the associative, idempotent, symmetric, and order-preserving operations on (finite) chains in terms of properties of (the Hasse diagram of) their associated semilattice order. In ... [more ▼] We characterize the associative, idempotent, symmetric, and order-preserving operations on (finite) chains in terms of properties of (the Hasse diagram of) their associated semilattice order. In particular, we prove that the number of associative, idempotent, symmetric, and order-preserving operations on an n-element chain is the nth Catalan number. [less ▲] Detailed reference viewed: 380 (78 UL)Conservative median algebras and semilattices ; Marichal, Jean-Luc ; Teheux, Bruno 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 ▲] Detailed reference viewed: 202 (13 UL)Hypomorphic Sperner systems and non-reconstructible functions Couceiro, Miguel ; Lehtonen, Erkko ; Schölzel, Karsten in Order: A Journal on the Theory of Ordered Sets and its Applications (2014) A reconstruction problem is formulated for Sperner systems, and infinite families of non-reconstructible Sperner systems are presented. This has an application to a reconstruction problem for functions of ... [more ▼] A reconstruction problem is formulated for Sperner systems, and infinite families of non-reconstructible Sperner systems are presented. This has an application to a reconstruction problem for functions of several arguments and identification minors. Sperner systems being representations of certain monotone functions, infinite families of non-reconstructible functions are thus obtained. The clones of Boolean functions are completely classified in regard to reconstructibility. [less ▲] Detailed reference viewed: 140 (29 UL)Parametrized arity gap Couceiro, Miguel ; Lehtonen, Erkko ; Waldhauser, Tamás 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 ▲] Detailed reference viewed: 53 (3 UL)On the Poset of Computation Rules for Nonassociative Calculus Couceiro, Miguel ; in Order: A Journal on the Theory of Ordered Sets and its Applications (2012), 30(1), 269-288 Detailed reference viewed: 85 (1 UL)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: 110 (1 UL)Associative polynomial functions over bounded distributive lattices Couceiro, Miguel ; Marichal, Jean-Luc 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 ▲] Detailed reference viewed: 116 (6 UL)On the homomorphism order of labeled posets ; Lehtonen, Erkko 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 ▲] Detailed reference viewed: 96 (2 UL)Descending chains and antichains of the unary, linear, and monotone subfunction relations Lehtonen, Erkko 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 ▲] Detailed reference viewed: 93 (4 UL)Aggregation on finite ordinal scales by scale independent functions Marichal, Jean-Luc ; 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 ▲] Detailed reference viewed: 102 (2 UL) |
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