Fuzzy Semiprime and Fuzzy Prime Subsets of Ordered Groupoids
Abstract
A fuzzy subset $f$ of an ordered
groupoid (or groupoid) $S$ is called fuzzy semiprime if $f(x)\ge
f(x^2)$ for every $x\in S$; it is called fuzzy prime if $f(xy)\le \min \{f(x),f(y)\}$ for every $x,y\in S$ (Definition 1). Following the terminology of
semiprime subsets of ordered groupoids (groupoids) and the
terminology of ideal elements of $poe$-groupoids (: ordered
groupoids possessing a greatest element), a fuzzy subset $f$ of
an ordered groupoid (or groupoid) should be called fuzzy semiprime
if for every fuzzy subset $g$ of $S$ such that $g^2:=g\circ g\preceq f$, we have $g\preceq f$; it should be called prime if for any fuzzy subsets $h$, $g$ of $S$ such that $h\circ g\preceq f$ we have $h\preceq f$ of $g\preceq f$ (Definition 2). And this is because if $S$ is a groupoid or an ordered groupoid, then the set of all fuzzy subsets of $S$ is a $poe$-groupoid. What is the relation between these two definitions? that is between the Definition 1 (the usual definition we always use) and the Definition 2 given in this paper? The present paper gives the related answer
groupoid (or groupoid) $S$ is called fuzzy semiprime if $f(x)\ge
f(x^2)$ for every $x\in S$; it is called fuzzy prime if $f(xy)\le \min \{f(x),f(y)\}$ for every $x,y\in S$ (Definition 1). Following the terminology of
semiprime subsets of ordered groupoids (groupoids) and the
terminology of ideal elements of $poe$-groupoids (: ordered
groupoids possessing a greatest element), a fuzzy subset $f$ of
an ordered groupoid (or groupoid) should be called fuzzy semiprime
if for every fuzzy subset $g$ of $S$ such that $g^2:=g\circ g\preceq f$, we have $g\preceq f$; it should be called prime if for any fuzzy subsets $h$, $g$ of $S$ such that $h\circ g\preceq f$ we have $h\preceq f$ of $g\preceq f$ (Definition 2). And this is because if $S$ is a groupoid or an ordered groupoid, then the set of all fuzzy subsets of $S$ is a $poe$-groupoid. What is the relation between these two definitions? that is between the Definition 1 (the usual definition we always use) and the Definition 2 given in this paper? The present paper gives the related answer
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