论文标题
在含义的Zroupoid中，半分布和惠特曼财产
Semidistributivity and Whitman Property in Implication Zroupoids
论文作者
论文摘要
在2012年，第二作者介绍并研究了含义Zroupoids的品种$ \ MATHCAL {I} $，这些Zroupoids概括了De Morgan代数和$ \ lor $ -semilattices，并以$ 0 $ $ 0 $。一个代数$ \ mathbf a = \ langle a，\ to，0 \ rangle $，其中$ \ to $ to $ as binary as Binary ex and $ 0 $是一个常数，称为\ emph {Incemation zroupoid}（$ \ nathcal {$ \ nathcal {i} $（x \ to y）\ to z \ about [（z'\ to x）\ to（y \ to z）']''$，其中$ x'：= x \ to 0 $，$ 0''\ oft 0 $。 令$ \ mathcal {i} $表示含义的Zroupoids和$ \ Mathbf a \ in \ Mathcal {i} $。对于$ x，y \ in \ mathbf a $，令$ x \ land y：=（x \ to y'）'$和$ x \ lor y：=（x'\ land y'）'$。在较早的论文中，我们证明，如果$ \ mathbf a \ in \ mathcal {i} $，则代数$ \ mathbf a_ {mj} = \ langle a，\ lor，\ lor，\ land \ land \ langle $是Bisemigroup。在本文中，我们将半分布性的概念从晶格中概述到双元组，并证明，对于每一个$ \ mathbf a \ in \ mathcal {i} $，bisemigroup $ \ mathbf a_ {mj} $的概念是半分布的。其次，我们将惠特曼的财产从晶格概括为双元素，并证明$ \ mathcal i $的subvarietion $ \ mathcal {mej} $，由身份定义：$ x \ land y \ land y \ of x \ lor y $，满足惠特曼物业。
In 2012, the second author introduced and studied the variety $\mathcal{I}$ of implication zroupoids that generalize De Morgan algebras and $\lor$-semilattices with $0$. An algebra $\mathbf A = \langle A, \to, 0 \rangle$, where $\to$ is binary and $0$ is a constant, is called an \emph{implication zroupoid} ($\mathcal{I}$-zroupoid, for short) if $\mathbf A$ satisfies: $(x \to y) \to z \approx [(z' \to x) \to (y \to z)']'$, where $x' : = x \to 0$, and $ 0'' \approx 0$. Let $\mathcal{I}$ denote the variety of implication zroupoids and $\mathbf A \in \mathcal{I}$. For $x,y \in \mathbf A$, let $x \land y := (x \to y')'$ and $x \lor y := (x' \land y')'$. In an earlier paper we had proved that if $\mathbf A \in \mathcal{I}$, then the algebra $\mathbf A_{mj} = \langle A, \lor, \land \rangle$ is a bisemigroup. In this paper we generalize the notion of semi-distributivity from lattices to bisemigroups and prove that, for every $\mathbf A \in \mathcal{I}$, the bisemigroup $\mathbf A_{mj}$ is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety $\mathcal{MEJ}$ of $\mathcal I$, defined by the identity: $x \land y \approx x \lor y$, satisfies the Whitman Property.
