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1
Content available A Note on some Characterization of Distributive Lattices of Finite Length
100%
Łazarz M. , Siemieńczuk K. , Department of Logic and Methodology of Sciences University of Wrocław P. , lazarzmarcin@poczta.onet.pl , logika6@gmail.com
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nr 1-2
EN
Using known facts we give a simple characterization of the distributivity of lattices of finite length.
2
Distributive lattice of minimum cut sets of a directed graph
100%
Grishkevich A. A.
Informatyka Teoretyczna i Stosowana
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2004
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tom R. 4, nr 7
7-22
EN
The algebraic operations above cut sets of a digraph are entered. Is demonstrated, that the set of minimum cut sets separating two vertices of the graph, bas structure of a distributive lattice. The concept of a undecomposable (nonreducible) cut set is reviewed and the representation of set of minimum cut sets is obtained on the basis of a subset of undecomposable cut sets. The special collections of cut sets are established, in which one the nonreducible cut set is extreme. The obtained results are illustrated by example.
3
Content available remote Semiring of Sets
80%
Coghetto R.
Formalized Mathematics
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2014
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tom 22
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nr 1
79-84
EN
Schmets [22] has developed a measure theory from a generalized notion of a semiring of sets. Goguadze [15] has introduced another generalized notion of semiring of sets and proved that all known properties that semiring have according to the old definitions are preserved. We show that this two notions are almost equivalent. We note that Patriota [20] has defined this quasi-semiring. We propose the formalization of some properties developed by the authors.
4
Content available Bounded lattices with antitone involutions and properties of MV-algebras
80%
Chajda I. , Emanovský P.
Discussiones Mathematicae - General Algebra and Applications
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2004
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tom 24
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nr 1
31-42
EN
We introduce a bounded lattice L = (L;∧,∨,0,1), where for each p ∈ L there exists an antitone involution on the interval [p,1]. We show that there exists a binary operation · on L such that L is term equivalent to an algebra A(L) = (L;·,0) (the assigned algebra to L) and we characterize A(L) by simple axioms similar to that of Abbott's implication algebra. We define new operations ⊕ and ¬ on A(L) which satisfy some of the axioms of MV-algebra. Finally we show what properties must be satisfied by L or A(L) to obtain all axioms of MV-algebra.
5
Content available On some finite groupoids with distributive subgroupoid lattices
80%
Pióro K.
Discussiones Mathematicae - General Algebra and Applications
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2002
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tom 22
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nr 1
25-31
EN
The aim of the paper is to show that if S(G) is distributive, and also G satisfies some additional condition, then the union of any two subgroupoids of G is also a subgroupoid (intuitively, G has to be in some sense a unary algebra).
6
Content available Distributivity of bounded lattices with sectionally antitone involutions
80%
Chajda I.
Discussiones Mathematicae - General Algebra and Applications
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2005
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tom 25
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nr 2
155-163
EN
We present a simple condition under which a bounded lattice L with sectionally antitone involutions becomes an MV-algebra. In thiscase, L is distributive. However, we get a criterion characterizingdistributivity of L in terms of antitone involutions only.
7
Content available remote A Discrete Representation for Dicomplemented Lattices
80%
Düntsch I. , Kwuida L. , Orłowska E.
Fundamenta Informaticae
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2017
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tom Vol. 156, nr 3/4
281--295
EN
Dicomplemented lattices were introduced as an abstraction of Wille’s concept algebras which provided negations to a concept lattice. We prove a discrete representation theorem for the class of dicomplemented lattices. The theorem is based on a topology free version of Urquhart’s representation of general lattices.
8
Content available Prime ideals in the lattice of additive induced-hereditary graph properties
80%
Berger A. , Mihók P.
Discussiones Mathematicae Graph Theory
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2003
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tom 23
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nr 1
117-127
EN
An additive induced-hereditary property of graphs is any class of finite simple graphs which is closed under isomorphisms, disjoint unions and induced subgraphs. The set of all additive induced-hereditary properties of graphs, partially ordered by set inclusion, forms a completely distributive lattice. We introduce the notion of the join-decomposability number of a property and then we prove that the prime ideals of the lattice of all additive induced-hereditary properties are divided into two groups, determined either by a set of excluded join-irreducible properties or determined by a set of excluded properties with infinite join-decomposability number. We provide non-trivial examples of each type.
9
Content available remote Left-outtermost extension of some varieties
70%
Płonka J.
Demonstratio Mathematica
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2003
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tom Vol. 36, nr 1
37--51
EN
Let r : F - > N be a type of algebras F is a nonempty set of fundamental operation symbols and N is the set of all positive integers. An identity ip fa if) of type T we call left-outermost if the left-outermost variables in ip and ip are the same. For a variety V of type r we denote by Vi the variety of type r defined by all left-outermost identities from Id(V). Vl is called the left-outermost extension of V. In this paper we study minimal generics, subdirectly irreducible algebras and lattices of subvarieties in left-outermost extensions of some generalizations of the variety D of all distributive lattices.
10
Content available remote Multi-Valued MSO Logics Over Words and Trees
70%
Droste M. , Kuich W. , Rahonis G.
Fundamenta Informaticae
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2008
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tom Vol. 84, nr 3-4
305-327
EN
We introduce multi-valued Büchi and Muller automata over distributive lattices and a multi-valued MSO logic for infinite words. For this logic, we prove the expressive equivalence of w-recognizable and MSO-definable infinitary formal power series over distributive lattices with negation function. Then we consider multi-valued Muller tree automata and a multi-valued MSO logic for trees over distributive lattices. For this logic, we establish a version of Rabin's theorem for infinitary tree series.
11
Content available A reduction theorem for ring varieties whose subvariety lattice is distributive
70%
Volkov M.
Discussiones Mathematicae - General Algebra and Applications
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2010
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tom 30
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nr 1
119-132
EN
We prove a theorem (for arbitrary ring varieties and, in a stronger form, for varieties of associative rings) which basically reduces the problem of a description of varieties with distributive subvariety lattice to the case of algebras over a finite prime field.
12
Content available remote An Algebraic Foundation for Linguistic Reasoning
60%
Huynh. V.N. , Murai T. , Nakamori Y.
Fundamenta Informaticae
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2007
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tom Vol. 78, nr 2
271-294
EN
It is well known that algebraization has been successfully applied to classical and non-classical logics (Rasiowa and Sikorski, 1968). Following this direction, an ordered-based approach to the problem of finding out a tool to describe algebraic semantics of Zadeh's fuzzy logic has been introduced and developed by Nguyen Cat-Ho and colleagues during the last decades. In this line of research, RH_algebra has been introduced in [20] as a unified algebraic approach to the natural structure of linguistic domains of linguistic variables. It was shown that every RH_algebra of a linguistic variable with a chain of the primary terms is a distributive lattice. In this paper we will examine algebraic structures of RH_algebras corresponding to linguistic domains having exactly two distinct primary terms, one being an antonym of the other, called symmetrical RH_algebras. Computational results for the relatively pseudo-complement operation in these algebras will be given.
13
Content available Algorithm and program for finding minimal and quasi-minimal cuts in graphs
60%
Grishkeyich A. , Piątek Ł.
Zeszyty Naukowe Wydziału Elektrotechniki i Automatyki Politechniki Gdańskiej
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2008
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tom Nr 25
49-52
EN
It was found, that the set of minimum cuts, separating two chosen vertices in graph, have the structure of distributive lattice. It was developed an effective procedure for finding the set of 1, 2 and 3 elements cuts in graph based on the consideration of distributive lattice of the set of minimum cuts. The procedure consists of first, the algorithm for finding indecomposable minimal cuts of distributive lattice. Second, algorithm for synthesis, using resulting subset from stage one, of the entire set of minimum cuts. The third, is the algorithm for describing the set of quasi-minimum (close to the minimum, next to minimum) cuts in the form of sum of distributive lattices of minimal cuts found for the modified function of weight. The computer program, implementing these algorithms, is presented with examples.
PL
Ustalono, że zbiór minimalnych przekrojów, rozdzielających dwa zadane wierzchołki grafu, z wprowadzonymi na i operacjami ma strukturę kraty dystrybutywnej. Opracowano skuteczną algorytmiczną procedurę znajdowania zbioru jed dwu i trzy elementowych przekrojów grafu bazującą na rozpatrzeniu dystrybutywnych krat zbioru minimalnych przekrojów grafu. Procedura składa się z, po pierwsze, algorytmu szukania nierozkładalnych minimalnych przekrojów Dystrybutywnej, po drugie, algorytmu syntezy po tym podzbiorze w kracie dystrybutywnej całego poszukiwanego zbioru minimalnych przekrojów i, po trzecie, algorytmu opisu zbioru quasi-minimalnych (bliskich do minimalnych, następnych po minimalnych) przekrojów w formie sumy krat dystrybutywnych minimalnych przekrojów, znalezionych dla zmodyfikowanej funkcji wagi. Przedstawiono zrealizowany program komputerowy i przedstawiono przykłady pracy programu.
14
Content available Distributive lattices with a given skeleton
60%
Grygiel J.
Discussiones Mathematicae - General Algebra and Applications
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2004
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tom 24
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nr 1
75-94
EN
We present a construction of finite distributive lattices with a given skeleton. In the case of an H-irreducible skeleton K the construction provides all finite distributive lattices based on K, in particular the minimal one.
15
Content available remote Properties of differences in B-rings
51%
Dadej A. , Halik K.
Journal of Mathematics and Applications
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2011
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tom Vol. 34
5-13
EN
Motivated by Pettis' extensions of Sierpinski theorems on generated families of sets, we consider B-rings, a generalization of the notion of Boolean algebras, and present their various properties. In particular, we discuss properties of differences which will be used in the proofs of results given in our forthcoming papers.
16
Algorytm przeliczania dwuelementowych minimalnych przekrojów grafa skierowanego
51%
Grishkevich A. A.
Informatyka Teoretyczna i Stosowana
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2006
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tom R. 6, nr 10
85-100
PL
Zaproponowano sposób kodowania kraty dystrybutywnej minimalnych przekrojów grafu. Opracowano algorytm przeliczania minimalnych dwuelementowych przekrojów grafu w kracie dystrybutywnej minimalnych przekrojów. Zaprezentowano przykład działania algorytmu.
EN
Coding of the distributive lattice of the minimal cuts graphs is offered. Enumerating algorithm of the minimal two-element cuts of the graph in a distributive lattice of the minimal cuts is developed. The example of work of the algorithm is presented.
17
Content available Power-ordered sets
51%
Goldstern M. , Schweigert D.
Discussiones Mathematicae - General Algebra and Applications
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2002
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tom 22
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nr 1
39-46
EN
We define a natural ordering on the power set 𝔓(Q) of any finite partial order Q, and we characterize those partial orders Q for which 𝔓(Q) is a distributive lattice under that ordering.
18
Content available Finite orders and their minimal strict completion lattices
51%
Bordalo G. , Monjardet B.
Discussiones Mathematicae - General Algebra and Applications
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2003
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tom 23
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nr 2
85-100
EN
Whereas the Dedekind-MacNeille completion D(P) of a poset P is the minimal lattice L such that every element of L is a join of elements of P, the minimal strict completion D(P)∗ is the minimal lattice L such that the poset of join-irreducible elements of L is isomorphic to P. (These two completions are the same if every element of P is join-irreducible). In this paper we study lattices which are minimal strict completions of finite orders. Such lattices are in one-to-one correspondence with finite posets. Among other results we show that, for every finite poset P, D(P)∗ is always generated by its doubly-irreducible elements. Furthermore, we characterize the posets P for which D(P)∗ is a lower semimodular lattice and, equivalently, a modular lattice.
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