• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.305-320
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• 2013

Topological semilattices with path-connected intervals are special abstract convex spaces. In this paper, we obtain generalized KKM type theorems and their analytic formulations, maximal element theorems and collectively fixed point theorems on abstract convex spaces. We also apply them to topological semilattices with path-connected intervals, and obtain generalized forms of the results of Horvath and Ciscar, Luo, and Al-Homidan et al..

• Kyungpook Mathematical Journal
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• v.47 no.2
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• pp.227-237
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• 2007

We define weak distributive $n$-semilattices and $n$-lattices, using variants of the absorption law and those of the distributive law. From a weak distributive $n$-semilattice, we construct direct system of subalgebras which are weak distributive $n$-lattices and show that its direct limit is a reflection of the category $wDn$-SLatt of the weak distributive $n$-semilattices.

• Communications of the Korean Mathematical Society
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• v.26 no.3
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• pp.405-416
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• 2011

In this paper, we first introduce the concept of left strongly simple po-semigroups, then we discuss properties and characterizations nil-extensions of left strongly simple po-semigroups and semilattices of leftstrongly simple po-semigroups. Finally, we give some characterizations of the chain of left strongly simple po-semigroups.

• 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 $x_0{\in}S$.

• Honam Mathematical Journal
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• v.32 no.4
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• pp.625-632
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• 2010

The notions of commutative pre-logics and terminal sections are introduced. Characterizations of a commutative prelogic are provided. Properties of deductive systems in pre-logics which are upper semilattices are considered.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.311-324
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• 2010

We prove that a regular ordered semigroup S is left simple if and only if every intuitionistic fuzzy left ideal of S is a constant function. We also show that an ordered semigroup S is left (resp. right) regular if and only if for every intuitionistic fuzzy left(resp. right) ideal A = <$\mu_A$, $\gamma_A$> of S we have $\mu_A(a)\;=\;\mu_A(a^2)$, $\gamma_A(a)\;=\;\gamma_A(a^2)$ for every $a\;{\in}\;S$. Further, we characterize some semilattices of ordered semigroups in terms of intuitionistic fuzzy left(resp. right) ideals. In this respect, we prove that an ordered semigroup S is a semilattice of left (resp. right) simple semigroups if and only if for every intuitionistic fuzzy left (resp. right) ideal A = <$\mu_A$, $\gamma_A$> of S we have $\mu_A(a)\;=\;\mu_A(a^2)$, $\gamma_A(a)\;=\;\gamma_A(a^2)$ and $\mu_A(ab)\;=\;\mu_A(ba)$, $\gamma_A(ab)\;=\;\gamma_A(ba)$ for all a, $b\;{\in}\;S$.

• Bulletin of the Korean Mathematical Society
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• v.31 no.2
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• pp.187-192
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• 1994

The author extended the small properties of topological semilattices to that of regular semigroups [3]. In this paper, it could be shown that a semigroup S is almost regular if and only if over bar RL = over bar R.cap.L for every right ideal R and every left ideal L of S. Moreover, it has shown that the Bohr compactification of an almost regular semigroup is regular. Throughout, a semigroup will mean a topological semigroup which is a Hausdorff space together with a continuous associative multiplication. For a semigroup S, we denote E(S) by the set of all idempotents of S. An element x of a semigroup S is called regular if and only if x .mem. xSx. A semigroup S is termed regular if every element of S is regular. If x .mem. S is regular, then there exists an element y .mem S such that x xyx and y = yxy (y is called an inverse of x) If y is an inverse of x, then xy and yx are both idempotents but are not always equal. A semigroup S is termed recurrent( or almost pointwise periodic) at x .mem. S if and only if for any open set U about x, there is an integer p > 1 such that x$^{p}$ .mem.U.S is said to be recurrent (or almost periodic) if and only if S is recurrent at every x .mem. S. It is known that if x .mem. S is recurrent and .GAMMA.(x)=over bar {x,x$^{2}$,..,} is compact, then .GAMMA.(x) is a subgroup of S and hence x is a regular element of S.