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Title the Law $Beta$A on Trices (New Contact Points of Algebraic Systems The law $beta$A on Trices (New contact points of algebraic Title systems, logics, languages, and computer sciences) Author(s) 堀内, 清光 Citation 数理解析研究所講究録 (2015), 1964: 152-162 Issue Date 2015-10 URL http://hdl.handle.net/2433/224200 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University 数理解析研究所講究録 第 1964 巻 2015 年 152-162 152 The law $\beta A$ on Trices Kiyomitsu Horiuchi Faculty of Science and Engineering, Konan University Okamoto, Higashinada, Kobe 658-8501, Japan We can consider various adaptations of laws in bisemilattice to in triple-semilattice. The character might change greatly even if the form is almost similar. We introduce \the law $\beta A$ of a such configuration type. This law means saying, \The result by three different operations of two different elements is different"' The definition is simple. But, as it plays an important role on triple-semilattice. We prove that the distribution law must not coexist with it. The order from one of the operation of trice that satisfy the $\beta A$ is not linearly order. And we discuss the relation between the triangle natural law and $\beta A.$ 1 Preliminaries A semilattice $(S, *)$ is a set $S$ with a single binary, idempotent, commutative and associative operation $*$ . $a*a=a$ (idempotent) (1) $a*b=b*a$ $(c\sigma$mmutative) (2) $a*(b*c)=(a*b)*c$ (associative) (3) Under the relation defined by $a\leq_{*}b\Leftrightarrow a*b=b$ , any semilattice $(S, *)$ is a partially ordered set $(S, \leq_{*})$ . Let $A$ be a set. A bisemilattice $(A, *1, *2)$ is an algebra which has two semilattice operations, that is, $(A, *1)$ and $(A, *2)$ are semilattices, respectively. Hence, we can construct two ordered sets $(A, \leq_{1})$ and $(A, \leq_{2})$ . A lattice is a bisemilattice which satisfy the following: $[B]$ (absorption laws) $a*(a*b)=a$ (4) $a*(a*b)=a$ (5) for every $a,$ $b\in A.$ The following laws are satisfied from absorption laws. For the following laws, there may be a suitable name as well as absorption laws. But, it is shown with the 153 sign ( $[A]$ etc.) to avoid bringing preconception. We defined these in [5]. We know $b $[A]$ and $[E]$ are equivalent. Let $a,$ \in A.$ $[A]$ if and only if the following condition hold: $a\leq_{1}b$ and $a\leq_{2}b$ $\Rightarrow$ $a=b$ . (6) $[E]$ if and only if the following condition hold: $a*1b=a*2b \Rightarrow a=b$ . (7) Let $T$ be a set. A triple-semilattice $(T, *1, *2, *3)$ is an algebra which has three semilattice operations. That is, $(T, *1)$ , $(T, *2)$ and $(T, *3)$ are semilattices, respectively. We construct three ordered sets $(T, \leq_{1})$ , $(T, \leq_{2})$ and $(T, \leq_{3})$ . Next properties are adaptation from $[B],$ $[A]$ and $[E]$ . Let $a,$ $b\in T.$ $[*B]$ if and only if the following six conditions hold: $((a*b)*b)*3b=b$ (8) $((a*1b)*b)*b=b$ (9) $((a*2b)*b)*b=b$ (10) $((a*2b)*b)*b=b$ (11) $((a*b)*1b)*b=b$ (12) $((a*b)*2b)*b=b$ (13) for every $a,$ $b\in T.$ $[*A]$ if and only if the following condition hold: $a\leq_{1}b$ and $a\leq_{2}b$ and $a\leq_{3}b$ $\Rightarrow$ $a=b$ . (14) $[*E]$ if and only if the following condition hold: $a*1b=a*2b=a*3b \Rightarrow a=b$ . (15) In [3], we proposed the algebra system with three semilattice operations with $[*B]$ and we call this $[*B]$ roundabout-absorption laws(or $r$-absorption laws). The algebraic structure is called a trice. We know $[*A]$ and $[*E]$ are equivalent. And these are satisfied from $[*B]$ . In a word, on bisemilattice, the $[A]$ is one of a weak law that can be led from the absorption law. The $[*A]$ that adjusted to triple-semilattice was a weak law as for $[A]$ (in [5]). 154 Let $T$ be a triple-semilattice. We say that an ordered triple $(a, b, c)$ is in a triangular situation if $(a, b, c)$ have the following properties: $a*3b=c$ and $a*2C=b$ and $b*1\mathcal{C}=a$ . (16) We draw three figures and show the example of triple-semilattices. The set $\{a, b, c\}$ of triangular situation is one of trice. When it is shown in figure, it is Fig.1. We say that $T$ has the triangle constructive law if $T$ has the following properties: $(d*1e)*3(d*2e)=d*3e$ (17) $(d*1e)*2(d*3e)=d*2e$ (18) $(d*3e)*1(d*2e)=d*1e$ (19) for all $d,$ $e\in T$ . That is, the ordered triple $(d*1e, d*2e, d*3e)$ is a triangular situation. We say that $T$ has the triangle natural law if $T$ has the following properties: $b*1\mathcal{C}=a$ and $a*2C=b$ $\Rightarrow$ $a*3b=c$ (20) $c*2a=b$ and $b*3a=c$ $\Rightarrow$ $b*1C=a$ (21) $a*3b=c$ and $c*1b=a$ $\Rightarrow$ $c*2a=b$ . (22) for $a,$ $b,$ $c\in T.$ We say that $T$ has the distributive law if $T$ has the following six properties: $a*(b*c) = (a*1b)*(a*c)$ (23) $a*2(b*1c) = (a*2b)*1(a*2c)$ (24) $a*1(b*3c) = (a*1b)*3(a*1c)$ (25) $a*3(b*1c) = (a*3b)*1(a*3c)$ (26) $a*2(b*3c) = (a*2b)*3(a*2^{\mathcal{C})}$ (27) $a*3(b*2\mathcal{C}) = (a*3b)*2(a*3c)$ (28) for every $a,$ $b,$ $c\in T$ . We investigated distributive trices in [4]. $c$ $\leq_{1} \leq_{2} \leq_{3}$ Three figures show the same set. A little circle expresses an element. An element in the same position in figures is the same. We depict three orders in the set by arrows of figures. We suppose that arrowhead is larger than the other end. Figure 1: triangular situation 155 2 Definition of $\beta A$ Now, we can think about different law $[\beta A]$ on triple-semilattice. This $[\beta A]$ is related $T$ to the $[A]$ as well as $[*A]$ . But, it is a strong law and important. Let be a triple- semilattice and $a,$ $b\in T.$ $[\beta A]$ if and only if the following three conditions hold: $a\leq_{1}b$ and $a\leq_{2}b$ $\Rightarrow$ $a=b$ (29) $a\leq_{2}b$ and $a\leq_{3}b$ $\Rightarrow$ $a=b$ (30) $a\leq_{3}b$ and $a\leq_{1}b$ $\Rightarrow$ $a=b$ . (31) Next $[\beta E]$ means saying, \The result of three different operations is different" The $[\beta A]$ and the $[\beta E]$ are equivalent. $[\beta E]$ if and only if the following three conditions hold: $a*b=a*b \Rightarrow a=b$ (32) $a*b=a*b \Rightarrow a=b$ (33) $a*b=a*b \Rightarrow a=b$ . (34) Theorem 1 $[\beta A]$ and $[\beta E]$ are equivalent. Accurately, (29) and (32) are equiva- lent, (30) and (33) are equivalent and (31) and (34) are equivalent. Proof This proof is the simulation of Proposition 2 in [5]. At first, we prove that (29) imply (32). Let $a*1b=a*2b=c$ . Then, $a\leq 1c,$ $b\leq_{1}c,$ $a\leq_{2}c$ and $b\leq_{2}c.$ $b\leq_{1}c$ $b\leq_{2}c$ $b=c.$ Rom (29), $a\leq_{1}c$ and $a\leq_{2}c$ imply $a=c$ . Similarly, and imply Hence $a=c=b$. Conversely, we prove that (32) imply (29). let $a\leq_{1}b$ and $a\leq_{2}b.$ Then, $a*1b=b$ and $a*2b=b$ . Hence, $a*1b=a*2b$ . If (32) is satisfied, $a=b.$ Therefore, (29) and (32) are equivalent. Similarly, (30) and (33) are equivalent and (31) and (34) are equivalent. " \ After this, we omit square brackets of $[\beta A]$ . We will use \`the law $\beta A$ or $\beta A$ The $[*B]$ imply the $[*A]$ , that is, the $[*A]$ that adjusted to triple-semilattice was a weak law as for $[A]$ . However, the situation of $\beta A$ is different. There are trices $\beta A$ which does not have the $\beta A$ . We show that some typical trices doesn't have the in the following two sections. 3 The $\beta A$ and the distributive law $T$ $b,$ $c $a\leq_{1}b,$ $b\leq_{2}$ $a$ Lemma 1 Let be a trice and $a,$ \in T$ . If and $a*3b=c$ , then $c\leq 1b$ and $c\leq_{2}a.$ 156 Proof Rom $r$-absorption law, $((a*2b)*3b)*1b=b$ . From assumption $b\leq_{2}a,$ $((a*2b)*3b)*1b=(a*3b)*1b=c*1b$ . Hence $c*1b=b$ , that is, $c\leq_{1}b$ . Similarly, $c\leq_{2}a.$ $T$ Lemma 2 Let be a non-trivial trice (not one point trice). There exist $a,$ $b\in T$ $(a\neq b)$ and $m,$ $n\in\{1$ , 2, 3 $\}$ $(m\neq n)$ such that $a\leq_{m}b,$ $b\leq_{n}a.$ Proof Let $c\neq d(c, d\in T).$ Fkom $c\leq_{1^{\mathcal{C}*}1}d$ and $d\leq_{1^{C*}1}d,$ $c\neq c*1d$ or $d\neq c*1d.$ Suppose that $d\neq c*1d$ and $d\leq_{1}c*1d$, that is, $d<c*1d$ . From $r$-absorption law, $((c*1d)*2d)*3d=d$ . Hence $(c*1d)*2d\leq_{3}d$. It is clear that $d\leq_{2}(c*1d)*2d$ . If $(c*1d)*2d\neq d$, let $a=(c*1d)*2d,$ $b=d,$ $m=3andn=2$ . On the other hand, if $(c*1d)*2d=d,$ $c*1d\leq_{2}d$ .
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