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Intersections of Finitely Generated Maximal Partial Clones Couceiro, Miguel ; in Journal of Multiple-Valued Logic & Soft Computing (2012), 19(1-3), 85-94 Let A be a finite non-singleton set. For A ={0, 1} we show that the set of all self-dual monotonic partial functions is a not finitely generated partial clone on {0, 1} and that it contains a family of ... [more ▼] Let A be a finite non-singleton set. For A ={0, 1} we show that the set of all self-dual monotonic partial functions is a not finitely generated partial clone on {0, 1} and that it contains a family of partial subclones of continuum cardinality. Moreover, for |A| ≥ 3, we show that there are pairs of finitely generated maximal partial clones whose intersection is a not finitely generated partial clone on A. [less ▲] Detailed reference viewed: 17 (0 UL)Polynomial functions over bounded distributive lattices Couceiro, Miguel ; Marichal, Jean-Luc in Journal of Multiple-Valued Logic & Soft Computing (2012), 18(3-4), 247-256 Let $L$ be a bounded distributive lattice. We give several characterizations of those $L^n \to L$ mappings that are polynomial functions, i.e., functions which can be obtained from projections and ... [more ▼] Let $L$ be a bounded distributive lattice. We give several characterizations of those $L^n \to L$ mappings that are polynomial functions, i.e., functions which can be obtained from projections and constant functions using binary joins and meets. Moreover, we discuss the disjunctive normal form representations of these polynomial functions. [less ▲] Detailed reference viewed: 67 (12 UL)Counting the maximal partial clones on a finite set Schölzel, Karsten in Journal of Multiple-Valued Logic & Soft Computing (2012), 18(2), 153-165 We show that different coherent relations specify different maximal partial clones. Then we describe a computer program to find all coherent relations and thus all maximal partial clones on 4-element, 5 ... [more ▼] We show that different coherent relations specify different maximal partial clones. Then we describe a computer program to find all coherent relations and thus all maximal partial clones on 4-element, 5-element, and 6-element sets. [less ▲] Detailed reference viewed: 32 (1 UL)A Sheffer-criterion for partial 4-valued logic Schölzel, Karsten in Journal of Multiple-Valued Logic & Soft Computing (2012), 18(2), 167-199 We determine the minimal covering of maximal partial clones in 4-valued logic. That means a necessary and sufficient condition for partial Sheffer functions in 4-valued logic in terms of the maximal ... [more ▼] We determine the minimal covering of maximal partial clones in 4-valued logic. That means a necessary and sufficient condition for partial Sheffer functions in 4-valued logic in terms of the maximal partial clones is established. Furthermore several statements about members of the minimal covering of the maximal partial clones for any finite-valued logic are established. [less ▲] Detailed reference viewed: 51 (1 UL)Representations and characterizations of polynomial functions on chains Couceiro, Miguel ; Marichal, Jean-Luc in Journal of Multiple-Valued Logic & Soft Computing (2010), 16(1-2), 65-86 We are interested in representations and characterizations of lattice polynomial functions f: L^n --> L, where L is a given bounded distributive lattice. In companion papers [5,6], we investigated certain ... [more ▼] We are interested in representations and characterizations of lattice polynomial functions f: L^n --> L, where L is a given bounded distributive lattice. In companion papers [5,6], we investigated certain representations and provided various characterizations of these functions both as solutions of certain functional equations and in terms of necessary and sufficient conditions. In the present paper, we investigate these representations and characterizations in the special case when L is a chain, i.e., a totally ordered lattice. More precisely, we discuss representations of lattice polynomial functions given in terms of standard simplices and we present new axiomatizations of these functions by relaxing some of the conditions given in [5,6] and by considering further conditions, namely comonotonic minitivity and maxitivity. [less ▲] Detailed reference viewed: 50 (5 UL) |
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