• International Journal of Contents
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• v.7 no.4
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• pp.56-63
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• 2011

Quantum mechanics is a branch of physics for a mathematical description of the particle wave, and it is applied to information technology such as quantum computer, quantum information, quantum network and quantum cryptography, etc. In 1936, Garrett Birkhoff and John von Neumann introduced the logic of quantum mechanics (quantum logic) in order to investigate projections on a Hilbert space. As another type of quantum logic, orthomodular implication algebra was introduced by Chajda et al. This algebra has the logical implication as a binary operation. In pure mathematics, there are many algebras such as Hilbert algebras, implicative models, implication algebras and dual BCK-algebras (DBCK-algebras), which have the logical implication as a binary operation. In this paper, we introduce the definitions and some properties of those algebras and clarify the relations between those algebras. Also, we define the implicative poset which is a generalization of orthomodular implication algebras and DBCK-algebras, and research properties of this algebraic structure.

• Kyungpook Mathematical Journal
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• v.43 no.4
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• pp.539-545
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• 2003

We present a set of equations that axiomatizes the class of all lattice implication algebras. The construction is different from the one given in [7], and the proof is direct: i.e., it does not rely on results from outside the realm of the lattice implication algebras, such as the theory of BCK-algebras. Then we show that the lattice H implication algebras are nothing more than the familiar Boolean algebras. Finally we obtain some negative results for the embedding of lattice implication algebras into Boolean algebras.

• Journal of applied mathematics & informatics
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• v.33 no.5_6
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• pp.739-748
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• 2015

The notion of quasi-lattice implication algebras is a generalization of lattice implication algebras. In this paper, we give an optimized definition of quasi-lattice implication algebra and show that this algebra is a distributive lattice and that this algebra is a lattice implication algebra. Also, we define a congruence relation Φ_F induced by a filter F and show that every congruence relation on a quasi-lattice implication algebra is a congruence relation Φ_F induced by a filter F.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.13 no.3
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• pp.208-214
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• 2013

Bandler and Kohout investigated the solvability of fuzzy relation equations with inf-implication compositions in complete lattices. Perfilieva and Noskova investigated the solvability of fuzzy relation equations with inf-implication compositions in BL-algebras. In this paper, we investigate various solutions of fuzzy relation equations with inf-implication compositions in pseudo BL-algebras.

• Communications of the Korean Mathematical Society
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• v.18 no.4
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• pp.621-628
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• 2003

We investigate some related properties of fuzzy filters and fuzzy implicative filters in lattice implication algebras. We find a characterization of fuzzy filters and fuzzy implicative filters, and we discuss a relation between fuzzy filters and fuzzy implicative filters in lattice implication algebras. Also we give an extension theorem of fuzzy implicative filters.

• Bulletin of the Korean Mathematical Society
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• v.38 no.1
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• pp.191-195
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• 2001

In this paper, a simple axiom system of lattice implication algebras is presented, it is convenient for verifying whether an algebra of type (2,2,2,1,0,0) becomes a lattice implication algebra.

• Honam Mathematical Journal
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• v.31 no.3
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• pp.323-333
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• 2009

We introduce the concepts of a fuzzifying filter and a fuzzifying implicative filter in lattice implication algebras.

• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.385-392
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• 2003

Using a fuzzy filter, a new congruence relation induced by the fuzzy filter is given in lattice implication algebras, and some of their properties are investigated.

• Bulletin of the Korean Mathematical Society
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• v.34 no.2
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• pp.193-198
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• 1997

In order to research the logical system whose propositional value is given in a lattice, Y. Xu [4] proposed the concept of lattice implication algebras, and discussed their some properties in [3] and [4]. Y. Xu and K. Qin [5] introduced the notions of filter and implicative filter in a lattice implication algebra, and investigated their properties. In this paper, in the first place, we give an equivalent condition of a filter, and provide some equivalent conditions that a filter is an implicative filter in a lattice implication algebra. By using these results, we construct an extension property for implicative filter.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.177-189
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• 2005

We obtain characterizations of T-fuzzy (implicative) filters and some properties of T-product of fuzzy (implicative) filters in lattice implication algebras. We also establish the extension property for T-fuzzy implicative filters.