• Communications of the Korean Mathematical Society
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• v.10 no.2
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• pp.285-292
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• 1995

Some nonpath-connected orthomodular lattices are given : Every infinite direct product of othomodular lattices containing infinitely many non-Boolean factors is a nonpath-connected orthomodular lattice. The orthomodular lattice of all closed subspaces of an infinite dimensional Hilbert space is a nonpath-connected orthomodular lattice.

• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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• pp.845-856
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• 2009

A block of an orthomodular lattice L is a maximal Boolean subalgebra of L. A site is a subalgebra of an orthomodular lattice L of the form S = A $\cap$ B, where A and B are distinct blocks of L. An orthomodular lattice L is called with finite sites if |A $\cap$ B| < $\infty$ for all distinct blocks A, B of L. We prove that there exists a weakly path-connected orthomodular lattice with finite sites which is not path-connected and if L is an orthomodular lattice such that the height of the join-semilattice [ComL]$\vee$ generated by the commutators of L is finite, then L is pathconnected.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.513-519
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• 1997

We show that each orthomodular lattice containing only atomic nonpath-connected blocks is a full subalgebra of an irreducible path-connected orthomodular lattice and there is a path-connected orthomodualr lattice L containing a weakly path-connected full subalgebra C(x) for some element x in L.

• Communications of the Korean Mathematical Society
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• v.19 no.2
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• pp.197-204
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• 2004

Every orthomodular lattice satisfying the loop lemma is the direct product of a Boolean algebra and an irreducible path-connected orthomodular lattice.

• Journal of the Korean Mathematical Society
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• v.33 no.2
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• pp.217-225
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• 1996

An orthomodular lattice (abbreviated by OML) is an ortholattice L which satisfies the orthomodular law: if x $\leq$ y, then $y = x \vee (x' \wedge y)$ [5]. A Boolean algebra B is an ortholattice satisfying the distributive law : $x \vee (g \wedge z) = (x \vee y) \wedge (x \vee z) \forall x, y, z \in B$.

• Communications of the Korean Mathematical Society
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• v.18 no.2
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• pp.207-214
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• 2003

There exist two distinct Symmetric differences in a non Boolean orthomodular lattics. Let L be an orthomodular lattice. Then L is a Boolean algebra if and only if one symmetric difference is equal to the other. An orthomodular lattice L is Boolean if and only if one of two symmetric differences of L is associative.

• Communications of the Korean Mathematical Society
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• v.15 no.3
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• pp.511-519
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• 2000

If G is a strict generalized orthomodular lattice and H={I｜I=[0, $\chi$, $\chi$$\in$G}, then H is prime ideal of the Janowitz's hull J(G) of G. If f is the janowitz's embedding, then the set of all commutatiors of f(G) equals the set of all commutators of the Janowitz's hull J(G) of G. Let L be an OML. Then L J(G) for a strict GOML G if and only if ther exists a proper nonprincipal prime ideal G in L.

• Proceedings of the Korea Contents Association Conference
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• 2013.05a
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• pp.369-370
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• 2013

Orthomodular lattice is a mathematical description of quantum theory which is based on the family CS(H) of all closed subspaces of a Hilbert space H. A partial multiplier is a function F from a non-empty subset D of a commutative semigroup A into A such that F(x)y = xF(y) for every elements x, y in A. In this paper, we define the notion of multipliers on orthomodular lattices and give some properties of multipliers. Also, we characterize some properties of orthomodular lattices by multipliers.

• Journal of Convergence for Information Technology
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• v.7 no.3
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• pp.65-71
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• 2017

The quantum logic was introduced by G. Birkhoff and 1. von Neumann in order to study projections of a Hilbert space for a formulation of quantum mechanics, and Husimi proposed orthomodular law and orthomodular lattices to complement the quantum logic. Abott introduced orthoimplication algebras and its properties to investigate an implication of orthomodular lattice. The commuting relation is an important property on orthomodular lattice which is related with the distributive law and the modular law, etc. In this paper, we define a binary operation on orthoimplication algebra and the greatest lower bound by using this operation and research some properties of this operation. Also we define a homomorphism and characterize the commuting relation of orthoimplication algebra by the homomorphism.

• Bulletin of the Korean Mathematical Society
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• v.31 no.1
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• pp.61-72
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• 1994

Every irreducible block-finite orthomodular lattice is simple [9] and every irreducible orthomodular alttice such that no proper p-ideal of L contains infinitely many commutators is simple [5]. Every finite (height) OML L which does not belong to the varitety generated by MO2 has one of the OML MO3, 2$^{3}$.2$^{2}$, D$_{16}$ OMLHOUSE as the homomorpyhic image of a subalgebra of L [3]. In this paper, we extend these results.s.