• 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.

• Journal of the Chungcheong Mathematical Society
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• v.29 no.3
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• pp.491-502
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• 2016

In this paper, as a generalization of symmetric bi-derivations and symmetric bi-f-derivations of a lattice, we introduce the notion of symmetric bi-(f, g)-derivations of a lattice. Also, we define the isotone symmetric bi-(f, g)-derivation and obtain some interesting results about isotone. Using the notion of $Fix_a(L)$ and KerD, we give some characterization of symmetric bi-(f, g)-derivations in a lattice.

• Journal of the Korean Mathematical Society
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• v.15 no.1
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• pp.59-61
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• 1978

Let $X_1$, $X_2$, ${\cdots}$, $X_n$ be i.i.d. uniform (0,1) random variables. Let $f_n(x)$ denote the probability density function (p.d.f.) of $T_n={\sum}^n_{i=1}X_i$. Consider a set S(x ; ${\delta}$) of lattice points defined by S(x ; ${\delta}$) = $x{\mid}x={\delta}+j$, j=0, 1, ${\cdots}$, n-1, $0{\leq}{\delta}{\leq}1$} The lattice distribution induced by the p.d.f. of $T_n$ is defined as follow: (1) $f_n^{(\delta)}(x)=\{f_n(x)\;if\;x{\in}S(x;{\delta})\\0\;otherwise.$. In this paper we show that $f_n{^{(\delta)}}(x)$ is a probability function thus we obtain a family of lattice distributions {$f_n{^{(\delta)}}(x)$ : $0{\leq}{\delta}{\leq}1$}, that the mean and variance of the lattice distributions are independent of ${\delta}$.

• Korean Journal of Mathematics
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• v.27 no.4
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• pp.1119-1131
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• 2019

In this paper, we introduce the notion of generalized f-derivation of lattice implication algebra and investigate some related properties. Also, we prove that if D is a generalized f-derivation associated with an f-derivation d of L, then D(x → y) = f(x) → D(y) for all x, y ∈ L.

• Bulletin of the Korean Mathematical Society
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• v.45 no.4
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• pp.701-707
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• 2008

In this paper, as a generalization of derivation on a lattice, the notion of f-derivation for a lattice is introduced and some related properties are investigated.

• East Asian mathematical journal
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• v.36 no.1
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• pp.1-11
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• 2020

In this paper, we introduce the notion of symmetric bi-f-derivation of lattice implication algebra and investigated some related properties. Also, we prove that if D is a symmetric bi-f-derivation of L, then D(x → y, z) = f(x) → D(y, z) for all x, y, z ∈ L.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.2
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• pp.179-189
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• 2019

In this paper, we introduce the notion of symmetric bi-generalized derivation of lattice implication algebra L and investigated some related properties. Also, we prove that a map $F:L{\times}L{\rightarrow}L$ is a symmetric bi-generalized derivation associated with symmetric bi-derivation D on L if and only if F is a symmetric map and it satisfies $F(x{\rightarrow}y,z)=x{\rightarrow}F(y,z)$ for all $x,y,z{\in}L$.

• Journal of the Chungcheong Mathematical Society
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• v.35 no.2
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• pp.125-136
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• 2022

The goal of this paper is to introduce the notion of generalized symmetric bi-f-derivations in lattices and to study some properties of generalized symmetric f-derivations of lattice. Moreover, we consider generalized isotone symmetric bi-f-derivations and fixed sets related to generalized symmetric bi-f-derivations.

• Communications of the Korean Mathematical Society
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• v.29 no.1
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• pp.27-36
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• 2014

In this paper, we introduce the notion of f-derivations from a semilattice S to a lattice L, as a generalization of derivation and f-derivation of lattices. Also, we define the simple f-derivation from S to L, and research the properties of them and the conditions for a lattice L to be distributive. Finally, we prove that a distributive lattice L is isomorphic to the class $SD_f(S,L)$ of all simple f-derivations on S to L for every ${\wedge}$-homomorphism $f:S{\rightarrow}L$ such that $f(x_0){\vee}f(y_0)=1$ for some $x_0,y_0{\in}S$, in particular, $$L{\simeq_-}=SD_f(S,L)$$ for every ${\wedge}$-homomorphism $f:S{\rightarrow}L$ such that $f(x_0)=1$ for some $x_0{\in}S$.

• East Asian mathematical journal
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• v.28 no.5
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• pp.605-615
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• 2012

In this paper, we show that any definite lattice over $\mathbb{F}_q[x]$ is universal if and only if it is quaternary and its discriminant is of degree 2, where $ch(\mathbb{F}_q){\neq}2$. The Four Conjecture follows as an immediate consequence.