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1
Content available Free Modal Pseudocomplemented De Morgan Algebras
100%
Figallo A. V. , Oliva N. , Ziliani A.
Bulletin of the Section of Logic
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2018
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tom 47
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nr 2
EN
Modal pseudocomplemented De Morgan algebras (or mpM-algebras) were investigated in A. V. Figallo, N. Oliva, A. Ziliani, Modal pseudocomplemented De Morgan algebras, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 53, 1 (2014), pp. 65–79, and they constitute a proper subvariety of the variety of pseudocomplemented De Morgan algebras satisfying xΛ(∼x)* = (∼(xΛ(∼x)*))* studied by H. Sankappanavar in 1987. In this paper the study of these algebras is continued. More precisely, new characterizations of mpM-congruences are shown. In particular, one of them is determined by taking into account an implication operation which is defined on these algebras as weak implication. In addition, the finite mpM-algebras were considered and a factorization theorem of them is given. Finally, the structure of the free finitely generated mpM-algebras is obtained and a formula to compute its cardinal number in terms of the number of the free generators is established. For characterization of the finitely-generated free De Morgan algebras, free Boole-De Morgan algebras and free De Morgan quasilattices see: [16, 17, 18].
2
Content available remote On n × m-valued Łukasiewicz-Moisil algebras
100%
Sanza C.
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tom 6
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nr 3
372-383
EN
n×m-valued Łukasiewicz algebras with negation were introduced and investigated in [20, 22, 23]. These algebras constitute a non trivial generalization of n-valued Łukasiewicz-Moisil algebras and in what follows, we shall call them n×m-valued Łukasiewicz-Moisil algebras (or LM n×m -algebras). In this paper, the study of this new class of algebras is continued. More precisely, a topological duality for these algebras is described and a characterization of LM n×m -congruences in terms of special subsets of the associated space is shown. Besides, it is determined which of these subsets correspond to principal congruences. In addition, it is proved that the variety of LM n×m -algebras is a discriminator variety and as a consequence, certain properties of the congruences are obtained. Finally, the number of congruences of a finite LM n×m -algebra is computed.
3
Content available Congruences and Boolean filters of quasi-modular p-algebras
88%
El-Mohsen Badawy A. , Shum K.
Discussiones Mathematicae - General Algebra and Applications
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2014
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tom 34
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nr 1
109-123
EN
The concept of Boolean filters in p-algebras is introduced. Some properties of Boolean filters are studied. It is proved that the class of all Boolean filters BF(L) of a quasi-modular p-algebra L is a bounded distributive lattice. The Glivenko congruence Φ on a p-algebra L is defined by (x,y) ∈ Φ iff x** = y**. Boolean filters [Fₐ), a ∈ B(L) , generated by the Glivenko congruence classes Fₐ (where Fₐ is the congruence class [a]Φ) are described in a quasi-modular p-algebra L. We observe that the set $F_{B}(L) = {[Fₐ): a ∈ B(L)}$ is a Boolean algebra on its own. A one-one correspondence between the Boolean filters of a quasi-modular p-algebra L and the congruences in [Φ,∇] is established. Also some properties of congruences induced by the Boolean filters [Fₐ), a ∈ B(L) are derived. Finally, we consider some properties of congruences with respect to the direct products of Boolean filters.
4
Content available remote Bisimulation Cuts For Structuring Markov Transition Systems
88%
Doberkat E.-E.
Fundamenta Informaticae
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2016
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tom Vol. 149, nr 4
363--383
EN
In Universal Algebra the structure of congruences for algebraic systems is fairly well investigated, and the relationship to the structure of the underlying system proper is well known. We propose a first step into this direction for studying the structure of congruences for stochastic relations. A Galois connection to a certain class of Boolean σ-algebras is exploited, atoms and antiatoms are identified, and it is show that a σ-basis exists. These constructions are applied to the problem of finding bisimulation cuts of a congruence. It cuts the relation through a span of morphisms with a minimum of joint events.
5
Content available remote On the Accuracy of Rough Approximations of Regular Languages
88%
Torres G. M.
Fundamenta Informaticae
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2014
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tom Vol. 132, nr 4
533--545
EN
In this paper we attempt to measure the accuracy of approximations of regular languages by languages in +−varieties (as defined by Eilenberg). These approximations are upper approximations in the sense of Pawlak’s rough set theory with respect to congruences belonging to the variety of congruences corresponding to the given +−variety. In our approach, the accuracy of an approximation is measured by the relative density of the object language in the approximation language and the asymptotic behavior of this quotient. In particular, we apply our measures of accuracy to k-definite, reverse k-definite, i, j-definite and k-testable approximations.
6
Content available remote Congruences of Edge-bipartite Graphs with Applications to Grothendieck Group Recognition. [Part 1], Inflation Algorithm Revisited
75%
Mróz A.
Fundamenta Informaticae
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2016
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tom Vol. 146, nr 2
121--144
EN
We study edge-bipartite graphs (bigraphs), a class of signed graphs, by means of the inflation algorithm which relies on performing certain elementary transformations on a given bigraph Δ, or equivalently, on the associated integral quadratic form qΔ: Zn → Z, preserving Gram Z-congruence. The ideas are inspired by classical results of Ovsienko and recent studies of Simson started in [SIAM J. Discr. Math. 27 (2013), 827-854], concerning classifications of integral quadratic and bilinear forms, and their Coxeter spectral analysis. We provide few modifications of the inflation algorithm and new estimations of its complexity for positive and principal loop-free bigraphs. We discuss in a systematic way the behavior and computational aspects of inflation techniques. As one of the consequences we obtain relatively simple proofs of several interesting properties of quadratic forms and their roots, extending known facts. On the other hand, the results are a first step of a solution of a variant of Grothendieck group recognition, a difficult combinatorial problem arising in representation theory of finite dimensional algebras and their derived categories, which we discuss in Part II of this two parts article with the same main title.
7
Content available remote Numerical investigation of least prime primitive roots modulo a prime number
63%
Paszkiewicz A.
Biuletyn Wojskowej Akademii Technicznej
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1999
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tom Vol. 48, nr 10
41-52
EN
For each of the 105,097,564 primes p from the interval [3,2 31] there exists a primitive root h(p)
PL
Dla każdej spośród 105,007,564 liczb pierwszych p z przedziału [3,2 31] istnieje pierwiastek pierwotny h(p)
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