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Dichotomy on intervals of strong partial Boolean clones Schölzel, Karsten in Algebra Universalis (2015), 73(3), 347-368 The following result has been shown recently in the form of a dichotomy: For every total clone $C$ on $\2 := \{0,1\}$, the set $\intervalD{C}$ of all partial clones on $\2$ whose total component is $C ... [more ▼] The following result has been shown recently in the form of a dichotomy: For every total clone $C$ on $\2 := \{0,1\}$, the set $\intervalD{C}$ of all partial clones on $\2$ whose total component is $C$, is either finite or of continuum cardinality. In this paper we show that the dichotomy holds, even if only strong partial clones are considered, i.e., partial clones which are closed under taking subfunctions: For every total clone $C$ on $\2$, the set $\intervalStr{C}$ of all strong partial clones on $\2$ whose total component is $C$, is either finite or of continuum cardinality. [less ▲] Detailed reference viewed: 72 (13 UL)Galois theory for sets of operations closed under permutation, cylindrification and composition Couceiro, Miguel ; Lehtonen, Erkko in Algebra Universalis (2012), 67(3), 273-297 A set of operations on A is shown to be the set of linear term operations of some algebra on A if and only if it is closed under permutation of variables, addition of inessential variables, and ... [more ▼] A set of operations on A is shown to be the set of linear term operations of some algebra on A if and only if it is closed under permutation of variables, addition of inessential variables, and composition, and if it contains all projections. A Galois framework is introduced to describe the sets of operations that are closed under the operations mentioned above, not necessarily containing all projections. The dual objects of this Galois connection are systems of pointed multisets, and the Galois closed sets of dual objects are described accordingly. Moreover, the closure systems associated with this Galois connection are shown to be uncountable (even if the closed sets of operations are assumed to contain all projections). [less ▲] Detailed reference viewed: 78 (1 UL)Uniqueness of minimal coverings of maximal partial clones Schölzel, Karsten in Algebra Universalis (2011), 65(4), 393-420 A partial function f on a k-element set Ek is a partial Sheffer function if every partial function on Ek is definable in terms of f. Since this holds if and only if f belongs to no maximal partial clone ... [more ▼] A partial function f on a k-element set Ek is a partial Sheffer function if every partial function on Ek is definable in terms of f. Since this holds if and only if f belongs to no maximal partial clone on Ek, a characterization of partial Sheffer functions reduces to finding families of minimal coverings of maximal partial clones on Ek. We show that for each k ≥ 2, there exists a unique minimal covering. [less ▲] Detailed reference viewed: 40 (0 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: 16 (1 UL)Closed classes of functions, generalized constraints and clusters Lehtonen, Erkko in Algebra Universalis (2010), 63(2-3), 203-234 Classes of functions of several variables on arbitrary nonempty domains that are closed under permutation of variables and addition of dummy variables are characterized by generalized constraints, and ... [more ▼] Classes of functions of several variables on arbitrary nonempty domains that are closed under permutation of variables and addition of dummy variables are characterized by generalized constraints, and hereby Hellerstein's Galois theory of functions and generalized constraints is extended to infinite domains. Furthermore, classes of operations on arbitrary nonempty domains that are closed under permutation of variables, addition of dummy variables, and composition are characterized by clusters, and a Galois connection is established between operations and clusters. [less ▲] Detailed reference viewed: 34 (0 UL) |
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