Scientiae Mathematicae Vol. 2, No. 3(1999), 373{384 373 TRICE AND TWO DELEGATES OPERATION KIYOMITSU HORIUCHI Received Octob er 1, 1999 Abstract. The concept of \trice " was intro duced in [5] and [6] for the purp ose of using truth value of fuzzy sets. In this pap er, weintro duce the notion of trice and some of its mathematical prop erties. Moreover, we shall showsometypical concrete examples of trice including those having two delegates op erations. 1. Intro duction. A semilattice (S; ) is a set S with a single binary, idemp otent, com- mutative and asso ciative op eration . (1) a a = a (idempotent) (2) a b = b a (commutativ e) (3) a (b c)=(a b) c (associativ e) The following Prop osition is well-known (see [1] p.10). Prop osition 1. Let (S; ) b e a semilattice. Under the relation de ned by a b () a b = b; any semilattice (S; ) is a partially ordered set (S; ). A lattice has two op erations. Let L b e a lattice with two op erations _ (join) and ^ (meet). Then, (L; _) and (L; ^) are semilattices. We can construct two ordered sets (L; ) and (L; ), resp ectively,by using the Prop osition 1 (a b , a _ b = b and _ ^ _ a b , a ^ b = b). The dual of (L; )is(L; ). That is, the order is nothing but ^ _ ^ _ the reverse of the order . The reason why _ and ^ intro duce the reverse order is that ^ lattices satisfy the absorption law: (4) a _ (a ^ b)=a; a ^ (a _ b)= a (absor ption) So, when a b (that is, a _ b = b), wehave a ^ b = a ^ (a _ b)=a by (4). Hence b a. _ ^ Supp ose that there is an ob ject with a rop e on a line, and that we are able to pull the rop e from the right (see Fig. 1). Then, wecanmove the ob ject to the right direction, but not to the left direction. This situation is considered to b e irreversible. Next, see Fig. 2. By pulling the rop e from either right or left directions, we can move the ob ject to anypoint on the line. This situation is considered to b e reversible. From Prop osition 1, a semilattice is a set having one order, i.e. one direction, likein Fig. 1. A lattice is a set having two orders, i.e. two directions, likeinFig.2.Any ob ject on the line can b e moved to another arbitrary p oint on the line by pulling the ob ject to the 1991 Mathematics Subject Classi cation. 06A12, 20N10, 20M10. Key words and phrases. semilattice, trice, roundab out-absorption law, triangular situation, two dele- gates op eration. 374 K.HORIUCHI - - Figure 1. one dimensional irreversible - - Figure 2. one dimensional reversible p ositive or negative directions. 6 > HY H H H H J J J J - J J^ Figure 3. two dimensional reversible Supp ose that there is an ob ject not on a line but in a plane (two dimensional Euclidian space). If wewanttomove the ob ject to an arbitrary p oint in the plane, it is sucientto have three \suitable" directions to pull the ob ject, as shown in Fig. 3. If the three directions are not \suitable", as shown in Fig. 4, we cannot move the original ob ject to an arbitrarily chosen target p oint. Movements in three directions are not nec- essarily restricted to the two dimensional plane (see Fig. 5 right). Consider that a lattice corresp onds to the one dimensional reversible case (Fig. 2). What systems with three binary op erations (semilattice) corresp ond to the two dimensional reversible cases (Fig. 3)? 2. Roundab out-Absorption Law. For A a nonempty set and n a p ositiveinteger, let (A; ; ;:::; ) b e an algebra with n binary op erations, and (A; ) b e a semilattice for 1 2 n i every i 2f1; 2; :::; ng. Then, (A; ; ;:::; )iscalledan-semilattice (See [7 ]). 1 2 n In this pap er, we will deal mainly with triple-semilattice, (i.e. n =3). We denote each order on A by (5) a b () a b = b; i i resp ectively. De nition 1. Let S b e the symmetric group on f1; 2; :::ng. An algebra (A; ; ; :::; ) n 1 2 n has the n-roundab out-absorption law if it satis es the following n!identities: (6) ((((a b) b) b)::: b)= b: (1) (2) (3) (n) TRICE 375 6 CO > > C HY H f f H f C H H Hj ? = ? - Figure 4. two dimensional irreversible 6 6 f f H H H A H H H A H H Hj Hj - A A A AU Figure 5. three dimensional cases for all a; b 2 A and for all 2 S . n We remark that \n-roundab out-absorption law" have di erent name \absorptivelaw." And arlgebaic researches yielded interesting results. (See [7]). However, weadoptn- roundab out-absorption law from the image of Prop osition 2. Of cause, the 2-roundab out-absorption law is the absorption law. An algebra (A; ; ) 1 2 which satis es the 2-roundab out-absorption law is a lattice. The op erations and are 1 2 denoted by _ and ^. De nition 2. An algebra (A; ; ; )which satis es the 3-roundab out-absorption 1 2 3 law issaidtobeatrice. To simplify explanation, we often omit \3" of \3-roundab out- absorption law." The op erations , and will b e denoted by % , - and # . 1 2 3 1 2 3 The \trice" is a notion to corresp ond to \lattice." Let T b e a set. Weintro duce three orders into T , under the condition that all two elements of T have a least upp er b ound for each order. Then, we can construct the set into a triple-semilattice. Example 1. Let T b e a set which consists of six p oints. Weintro duce three orders in the set by arrows of Fig. 6. Supp ose that arrowhead is larger than the other end (e.g. d e). 2 Then, T is a trice. That is, T has the 3-roundab out-absorption law. 1 2 3 c - c - c c c c c c c J J J] J] J^ J^ - J J c c c c c c e d J J] J^ J c c c c Figure 6. Example 1 376 K.HORIUCHI Example 2. Denote by R the set of all real numbers. Weintro duce three orders in the 2 two dimensional Euclidian space R as follows : p x y () x y and 0 y x 3(y x ) 1 1 1 2 2 1 1 p x y () x y and 0 y x 3(x y ) 2 1 1 2 2 1 1 p 3 jy x j: x y () x y 1 1 3 2 2 2 2 for x =(x ;x );y=(y ;y ) 2 R . This (R ; % ; - ; # )isatrice. Needless to say, 1 2 1 2 1 2 3 2 wecanmake another trice on R . De nition 3. Let (T; % ; - ; # ) b e a triple-semilattice. A sub-triple-semilatti ce 1 2 3 of T is a subset S of T such that (7) a; b 2 S impl y a % b; a - b; a # b 2 S: 1 2 3 If T is a trice, then a sub-triple-semilattice S is a trice. Wesay that S is a subtrice of T . Let T b e a trice. The emptysubsetof T is a subtrice. For any a 2 T ,onepoint set fag is a subtrice of T .Inabove Example 1, the set fc, d, eg isasubtriceofT .Wecanembed Example 1 in Example 2, as a subtrice. We can easily check that a lattice L has following prop erties: 8a; b 2 L 9c 2 Ls:t:a c b _ ^ 8a; b 2 L 9c 2 L s:t: a c b: ^ _ Prop osition 2. Let T b e a trice. For every a; b 2 T , there exist c; d 2 T suchthat a c d b. 1 2 3 Pro of Obviously, a a % b (a % b) - b. From ((a % b) - b) # b = b 1 1 2 1 2 1 2 3 (the roundab out-absorption law), (a % b) - b b. Let c = a % b and 1 2 3 1 d =(a % b) - b. Then, wehave a c d b. This completes the pro of. 1 2 1 2 3 One can prove the next Prop osition in a similary way. Prop osition 3. Let an algebra (A; ; ;:::; )have the n-roundab out-absorption law. 1 2 n For every a; b 2 A and for evry 2 S , there exists c ;:::;c 2 A such that n 1 n1 (8) a c c ::: c b: 1 2 n1 (1) (2) (3) (n1) (n) De nition 4. Wesay that (A; ; ;:::; )is attainable if (8) is true for every a; b 2 A. 1 2 n The \attainablity" of lattice corresp onds to the notion of reversiblity in the case of one dimension (Fig. 2). We can consider that the \attainablity" of trice corresp onds to the case of Fig. 3, that is, the two dimensional reversible cases. 1 2 3 c c c - - c c c c c c * HY H J J] ? b c H c c J^ * HY c c J H c c c c c d J J] ? J^ c J c c a Figure 7. Example 3 TRICE 377 Example 3.