A criterion for positive completeness in ternary logic

Марченков, С. С.

doi:10.1134/s1990478907040114

Cited by 4 publications

(12 citation statements)

References 3 publications

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“…With the use of the functions g 0b we obtain the remaining constants 1; 2; : : : ; .m C 1/p 1. Since the constants .m C 1/p; : : : ; k 1 belong to the class PG.g/ and the system of all constants is positively complete in the class P k [6], we conclude that the system of functions PG.g/ [ ff g is also positively complete in the class P k .…”

Section: Polgrendq/// D Qmentioning

confidence: 92%

“…If g is an idempotent function differing from the identity function, then the positive precompleteness of the class PG.g/ is established in [8]. If g is a permutation on E k which admits a decomposition into a product of cycles of the same prime length, then the positive precompleteness of the class PG.g/ is proved in [6].…”

Section: Polgrendq/// D Qmentioning

confidence: 96%

“…Indeed, as shown in [6], any positively closed class in P k is positively generated by the set of all its functions depending on k variables. So, if Q is an infinite set of functions, for an appropriate finite subset Q 0 Q the relation PosOEQ 0 D PosOEQ holds.…”

mentioning

confidence: 99%

“…The first of them is the positive closure operator (which is even more strong than the parametric closure operator). The positive closure operator has been investigated in [4][5][6][7][8][9]. In particular, it has been seen (see [4]) that there are precisely six positively closed classes of Boolean functions and that for any k 3 the number of positively closed classes in P k is strictly less than the number of subsemigroups with unity in the symmetric semigroup of degree k.…”

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confidence: 99%

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Definition of positively closed classes by endomorphism semigroups

Марченков¹

2012

Discrete Mathematics and Applications

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The positively closed classes of multivalued logic are characterised with the use of endomorphism semigroups. All subsemigroups of the symmetric semigroup of degree k are described which are semigroups of endomorphisms of positively closed classes in P k . On the base of the obtained results, we find all positively precomplete classes in P k .Prominent among a number of ways to classify the functions of multivalued logic are those based on the closure operators. On the set P k of the functions of k-valued logic, some operator O is defined; then the sets of functions in P k closed with respect to the operator O (usually referred to as the O-closed classes) form the O-classification of the set P k .The most known classification of such a kind is the classification based on the superposition operator. This classification has long been in use. But there exists an objective difficulty in its use: for any k 3 the number of closed classes in P k is continual (see [10]). In this connection, a great body of 'strong' closure operators have been introduced which for any k induce either finite or countable classifications on the set P k .One of the first strong closure operators is the parametric closure operator introduced by Kuznetsov in [3]. All twenty five parametrically closed classes of Boolean functions are found in [3] (see also [5]); the finiteness of the number of parametrically closed classes in P 3 is proved in [2], and for P k with k 4 it is established in [11]. It is worth noticing that for k 3 there is no complete description of parametrically closed classes in P k yet.Later, a more formalised definition of the parametric closure operator from the logic function viewpoint has been suggested in [4] (see also [5]). This definition permits to introduce more new strong closure operators. The first of them is the positive closure operator (which is even more strong than the parametric closure operator). The

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Section: Polgrendq/// D Qmentioning

confidence: 92%

Section: Polgrendq/// D Qmentioning

confidence: 96%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Definition of positively closed classes by endomorphism semigroups

Марченков¹

2012

Discrete Mathematics and Applications

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“…Note that the positive precompleteness of certain classes was proved in [3,4]. All the positively precom plete classes in P 2 and P 3 were found in [2,3].…”

Section: Pos[q] If and Only If End(q) ⊆ End( F )mentioning

confidence: 93%

Operator of positive closure

Марченков

2012

Dokl. Math.

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Positively closed classes of three-valued logic generated by one-place functions

Марченков¹

2009

Discrete Mathematics and Applications

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We consider the positive closure operator on the set P 3 of functions of three-valued logic. It is shown that in P 3 there are exactly 51 positively closed classes positively generated by one-place functions, 26 of these classes are generated by a single one-place function (including one of the positively precomplete in P 3 classes), and 25 classes are generated by two functions (including the class P 3 and the remaining 9 positively precomplete classes).

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A criterion for positive completeness in ternary logic

Cited by 4 publications

References 3 publications

Definition of positively closed classes by endomorphism semigroups

Definition of positively closed classes by endomorphism semigroups

Operator of positive closure

Positively closed classes of three-valued logic generated by one-place functions

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