数理解析研究所講究録
第 1846 巻 2013 年 19-26

19Homomorphism on Triple-semilattice
Kiyomitsu Horiuchi(堀内清光)
Faculty of Science and Engineering, Konan University(甲南大学理工学部)
Okamoto, Higashinada, Kobe 658-8501, Japan

Homomorphism is a fundamental concept in the algebra system with the operation.
In case of the algebra system with two or more operations, the concept of individual homomorphism can be considered respectively for each operation. For instance,
there are two operations, meet and join V, on the lattice. Hence, there are two homomorphism, -homomorphism, and -homomorphism. These two homomorphism
is independent for a general lattice. That is, there are mappings that is nothomomorphism but -homomorphism, and vice versa. In case of the totally orderedset, -homomorphism and -homomorphism are corresponding. In another algebra
system with two operations or more, there are some cases where the second operation
is sure to become homomorphism if the first operation is homomorphism. Moreover,another operation might become homomorphism from the homomorphism of two
operations or more.
We would like to introduce the order on sets of the operations of the algebra
system by using this. In this paper, it study the order on the mathematics object
known well “set operations” and “addition and multiplication on the set real number
or rational number field”. And it reports on the condition that the homomorphismof one operation of the remainder is approved by homomorphism in two operations
on triple-semilattice with three operations.
$\wedge$$\wedge$$\vee$$\wedge-$$\vee$$\wedge$

1$\vee$Homomorphic order

We adopt the notation in [1] for universal algebra. Suppose and are two algebrasof the same type $F.$
mapping : $Aarrow B$ is called a homomorphism from toif
$A$

$A$

$B$

$A$

$\alpha$(1)$\alpha f^{A}(a_{1}\ldots, a_{n})=f^{B}(\alpha a_{1}\ldots, \alpha a_{n})$for each -ary operation
$n$

$f$

in $F$ and each sequence

For individual operation in $F$ , amappingfrom to if
$f$

$A$

$\alpha$$B$$a_{1}\ldots,$ $a_{n}$from $A.$: $Aarrow B$ is called a -homomorphism
$f$

$B$

$\alpha f^{A}(a_{1}\ldots, a_{n})=f^{B}(\alpha a_{1}\ldots, \alpha a_{n})$.

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Definition 1
Let and be operations in $F$ . We say that is homomorphic, if every -homomorphism mapping isstronger than , and we writeof type $F$ and for every
every
algebras
andtwo-homomorphism, i.e., for: $Aarrow B$ if is -homomorphism then is -homomorphism. And we write, if. This order is called homomorphic order.
and$f_{1}$$f_{2}$$f_{1}$$f_{1}\gg f_{2}$$f_{2}$$f_{1}$$A$$f_{2}$$f_{1}$$\alpha$$\alpha$$f_{1}$$f_{2}$$\alpha$$f_{2}\gg f_{1}$$f_{1}\gg f_{2}$$\equiv f_{2}$$B$becomes homomorphism
In the definition, the set of mappinngs that operation
contain the set of mappinngs that operation becomes homomorphism. The weakest
operations are homomorphism on all mappings. Actually, projection $(f(a_{1}, a_{2})=a_{1})$is one of the weakest operations.
$f_{2}$$f_{1}$. Then, $\gg-1.$
Example 1 On the algebra of group, there are oprations an$Aarrow
B$is -homomorp hismThat is, for group $<A,$ $-1,1>and<B,$ $-1,1>$ , if :-homomorphism.
then i$d^{-1}$$\alpha$$\alpha$$\cdot$$\cdot,$$\cdot,$$s^{}$

Example 2 On the algebra of totally ordered set, we consider opemtions $\wedge and$$\vee of$ lattice.
. That is, for totally ordered $set<A,$ $\vee>and$
Then,$Aarrow
Bis\wedge-homomo7$-homomorphism and if$<B,$
phism then, if :-homomorphism.
-homomorp hism then$\wedge\equiv\vee$$\wedge,$$\vee>$$\wedge,$$\alpha is\vee$$\alpha$$\alpha is\vee$$\alpha is\wedge$

In case of a general lattice, it is natural that -homomorphism and -homomorphism
are independence. This order is more interesting when thinking adding the operation
made by the combination.
$\vee$$\wedge$$\vee and’$ . In addition to,
Example 3 On the Boolean algebm, there are opemtions$\vee and’$ can be represented
we consider Sheffer operation $a|b=(a\wedge b)’$ . Then,by . That is, $a\wedge b=(a|b)|(a|b),$ $a\vee b=(a|a)|(b|b)$ and $a’=a|a$ . Hence, $|\gg\vee,$$\wedge,$$|,$$\wedge,$$|$$\gg\wedge$

and$|\gg\prime.$

be operations in $F$ . We writeLetDefinition 2
if for every two algebras andof type $F$ and for every : $Aarrow B$ if-homomorphism for $k\in\{1, \ldots n\}$ then is -homomorphism.
$f_{0},$$f_{1},$$\{f_{1}, f_{2}, \ldots, f_{n}\}\gg$$f_{n}$$\ldots,$

$B$

$A$

$f_{0}$$f_{k}$$\alpha$In Example 3, it is clear that2$\alpha$$\alpha$is$f_{0}$$\{\vee, \prime\}\gg|.$

Observation of set operations

In this section, we study the order on set operation. This is a typical model of Boolean
algebra. Let $X$ and $Y$ be sets. The collection of all subsets of a set $X$ denoted by. The set operation usually used is enumerated as follows.

$\mathcal{P}(X)$nullary operation
unary operation
binary operation(emptyset) $X$ (whole set)$-($ complement)
difference)
intersection)(umion)$arrow($ implication)(Sheffer operation)$\phi$$\cup$$\cap($$\backslash ($$|$21$A\cup B=\{x\in A or x\in B\},$$\overline{A}=\{x\not\in A\},$$A\backslash B=\{x\in A$and$x\not\in B\},$$A\cap B=\{x\in A$ and $x\in B\},$$Aarrow B=\{x\not\in A or x\in B\},$$A|B=\{x\not\in A or x\not\in B\}.$

It is possible to compose the other binary operations. However, it is enough in whatenumerated in the above. (Is it possible to create essential and substantial ternary
operation onIt thinks about the condition whose mapping :is homomorphism
for these operations. And, the homomorphic order by the combination of these
operations is shown in Figure 1.
$\mathcal{P}(X)?)$$\alpha$$\mathcal{P}(X)arrow \mathcal{P}(Y)$Figure 1: The homomorphic order$\gg$$1=\{\cup, \cap, \phi, X, \backslash )arrow, -\}\equiv\{\cup, \cap, \phi, X\}\equiv\{\cap, -\}\equiv\{X, \backslash \}\equiv|$etc.

$2=\{\cup, \cap, \phi\}\equiv\{\cup, \backslash \}$$3=\{\cup, \phi, X\}$
$4=\{\cup, \cap, X\}\equiv\{\cap, arrow\}$$5=\{\cap, \phi, X\}$

$6=\{\phi, X, -\}\equiv\{\phi, -\}\equiv\{X, -\}$
$7=\{\cup, \cap\}$$8=\{\cup, \phi\}$$9=\{\cap, \phi\}$$10=\{\cup, X\}$

$11=\{\cap, X\}$

$12=\{\phi, X\}$

In this figure, we will pay attention torepresented byand . To similar,$U,$$\cap$$\phi$$\{\cup, \cap, \phi\}\gg\backslash$$-$. It is because thatcan not represented by$U,$$\cap,$$\phi$$\backslash$

can not

and $X$ . It

22is understood that representable and the homomorphic order are different concepts
even if in Boolean algebra. In Example 1 of the previous section, thecan not$-1$

represented by$\cdot.$Observation of operations on $R$

3

Let $R$ be the set of real numbers and be the set of rational numbers. In this section
we discuss the condition that a mapping from $R$ to $R$ becomes homomorphism forindividual operation in $R.$
First, $R$ is a totally ordered set, we have the operations and . If : $Rarrow R$ is-homomorphism $(that is, \alpha(x\vee y)=\alpha(x)\vee\alpha(y))$ , then is a monotone increase$(x\leq y$ imply.
The constant function on $R$ will be written as $c_{r}(x)=r(r\in R)$ . Although these
are unary operations in $R$ seemingly, it is actually these are nullary operations.
If is -homomorphism $(that is, \alpha(c_{v}(x))=c_{\tau}(\alpha(x)))$ , then $\alpha(r)=r.$
We define $f_{r}(r\in R)$ by $f_{r}(x)=rx$ . When $r=0,$ $f_{0}=c_{0}$ . For any mapping
ishomomorphism. If : $Rarrow R$ is -homomorphism $(that is, \alpha(f_{r}(x))=f_{r}(\alpha(x)))$ ,then $\alpha(r^{m}k)=r^{m}\alpha(k)$ for ever $m\in Z$ (the set of all integers) for every $k\in$ R. The
following consists of this.
$Q$$\alpha$$f$$\vee$$\vee$$\wedge$$\alpha$$\alpha$$\alpha(x)\leq\alpha(y))$$c_{7}(x)$$\alpha$$c_{r}$$\alpha,$$f_{1}$$f_{r}$$\alpha$Proposition 1
Let $r\in R$ such that $s=r^{n}(n\in Z)$ . If
homomorphism then is -homomorphism. That is,$s,$$\alpha$$\alpha$: $Rarrow R$ is$f_{r^{-}}$$f_{r}\gg f_{\theta}.$$f_{\epsilon}$For example,. However. Inaddition, if : $Rarrow R$ satisfies both-homomorphism and -homomorphism,$\alpha((-2)^{m}*3^{n}k)=(-2)^{m}*3^{n}\alpha(k)$thenfor ever $m,$ $n\in Z$ for every $k\in$ R. Hence,$\{f_{-2}, f_{3}\}\gg f_{(-2)^{n}*3^{n}}(m, n\in Z)$ . It is noted that the $\{(-2)^{m}*3^{n}|m, n\in Z\}$ is dense$f_{\sqrt{2}}\gg f_{2}\gg f_{4}\gg f_{8}\gg\ldots\gg f_{0}\gg f_{1},$$f_{-2}\gg f_{4},$$f_{4}\gg f_{2}$$f_{2}\equiv f_{2}-1$$f_{-2}$$\alpha$$f_{3}$

in $R.$

Next, we consider binary operation in R. The equation $\alpha(x+y)=\alpha(x)+\alpha(y)$is certainly Cauchy equation! This imply $\alpha(qx)=q\alpha(x)(q\in Q)$ , hence $+\gg$$\{f_{q}|q\in Q\}$ . But $\{f_{f}|r\in Q\}\#+$ . It is easy that $\{f_{f}|r\in R\}\gg+$ . However,fromthe existence of Hamel base that relates to the axiom of choice, $+*\{f_{r}|r\in R\}.$$+$

Proposition 2

For a mapping$\alpha$: $Rarrow R,$(3)$\{f_{r}|r\in R\}\gg+\gg\{f_{q}|q\in Q\}$If it thinks$+$

to be an operation in $Q$ and$\alpha$

to be a mapping from

$Q$

to$Q$

, then$+$$\equiv\{f_{q}|q\in Q\}.$

The situation changes completely if the condition of -homomorpism is added to$\vee$$\alpha:Rarrow R.$

Proposition 3

For a mapping$\alpha$: $Rarrow R,$$\{f_{-2}, f_{3}, \vee\}\equiv+$

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Moreover, we consider binary operation . If it thinks to be an operation in $Q$
and to be a mapping from $Q$ to $Q,$ -homomorpism and -homomorpism imply$c_{q}(x)=q$ for all $q\in$ Q. That is, $\{c_{1}, +\}\gg*.$ $If*$ is an operation in $R$ andis
a mapping $Rarrow R$ , then. However, the following are derived from theproposition above.
$*$$\alpha$$*$$+$$c_{1}$$\alpha$$\{c_{1}, +\}\ovalbox{\tt\small REJECT}*$For a mapping

Proposition 4: $Rarrow R,$$\alpha$(5)$\{c_{1}, +, \vee\}\equiv\{c_{1}, f_{-2}, f_{3}, \vee\}\gg*$

Let $n\in N$ and $n\neq 1$ . We define$x_{1}+x_{2}+\ldots+x_{n}$ . Then,
Proposition 5

For a mapping$t_{n}$-ary operation on $R$ by$n$$t_{n}(x_{1}, x_{2}, \ldots,x_{n})=$: $Rarrow R,$ $n\in N$ and $n\neq 1,$$\alpha$(6)$+\equiv t_{n}$It thinks about the operation of another type (average). Let $n\in N$ and $n\neq 1.$-ary operation on $R$ by $h_{n}(x_{1}, x_{2}, \ldots,x_{n})=(x_{1}+x_{2}+\ldots+x_{n})/n$ . Let :We define$h_{n}$$n$$\alpha$be -homomorphism, i.e.,.
We define $\beta(x)=\alpha(x)-\alpha(O)$ , then $\beta(x_{1}+x_{2}+\ldots+x_{n})=\beta(x_{1})+\beta(x_{2})+\ldots+\beta(x_{n})$ .
As is -homomorphism for $\alpha(0)=0,$ $+\equiv t_{n}\gg h_{n}$ . This fact is clear also in theanother way. From. Since
the Proposition 1 and 5, $t_{n}\equiv+\gg f_{n}\equiv f_{1/n}$ . Hence. But,
when $\alpha(x)=1,$ is not -homomorpism but -homomorpism.
$Rarrow R$$\alpha$$h_{n}$$\alpha(h_{n}(x_{1}, x_{2}, \ldots,x_{n}))=h_{n}(\alpha(x_{1}), \alpha(x_{2}), \ldots, \alpha(x_{n}))$$t_{n}$$h_{n}(x_{1}, x_{2}, \ldots, x_{n})=f_{1/n}(t_{n}(x_{1}, x_{2}, \ldots, x_{n})),$ $\{f_{1/n}, t_{n}\}\gg h_{n}$$t_{n}\equiv\{f_{1/n}, t_{n}\}\gg h_{n}$$\alpha$Proposition 6$+$$h_{n}$For a mapping$\alpha$: $Rarrow R,$ $n\in N$ and $n\neq 1,$(7)$+\gg h_{n}$We remark that the construction of the operation cannot be done fromalone.
The following proposition can be easily shown.
$h_{n}$

Proposition 7

For a mapping$\alpha$$+$operation: $Rarrow R,$ $n,$ $n’\in N$ and $n\neq 1\neq n’,$$h_{n}\equiv h_{n’}$(8)

It is clear that $\{f_{n}, h_{n}\}\gg t_{n}$ . Hence,

Proposition 8 For a mapping: $Rarrow R$, if there exists $n_{1}\in N(n_{1}\neq 1)$
such that is-homomorpism and there exists $n_{2}\in N(n_{2}\neq 1)$ such that is-homomorpism, thenis -homomorpism.
$\alpha$$\alpha$$h_{n_{2}}$$f_{n1}$$\alpha$$\alpha$$+$244

Observation of triple-semilattice

We proposed the algebra system with three operations in [4], [5]. It is a kind of
dimensional expansion of the lattice theory. The definition is written again here.
A semilattice $(S, *)$ is a set
associative operation$S$with a single binary, idempotent, commutative and$*.$

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(idempotent)(cammutative)(associative)$a*(b*c)=(a*b)*c$$a*a=a$$a*b=b*a$

be a set. Let $*1,$
andbe three binaarry operations
Definition 3
Leton . If $(T, *1),$ $(T, *2)$ and $(T, *3)$ are semilattices respectively, the quartet$(T, *1, *2, *3)$ is called a triple-semilattice.
$T$

$*3$

$*2$

$T$

The role that looks like the lattice in putting the following relational expression
is done though three operations are various the way things are going.
Definition 4Let
absorption laws.

$T$be a triple-semilattice. We call next six equality roundabout-

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(15)
(16)
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$((a*1b)*2b)*3b=b$$((a*1b)*3b)*2b=b$$((a*2b)*1b)*3b=b$$((a*2b)*3b)*1b=b$$((a*3b)*1b)*2b=b$$((a*3b)*2b)*1b=b$for every $b\in T$ . We say that have the roundabout-absorption laws if and only
if these six identity holds in it. triple-semilattice is called a trice if it satisfies theroundabout-absorption laws.
$a,$

$T$

$A$

Let $(T, *1, *2, *3)$ be a triplesemilattice and

$a,$

$b,$

$c\in T(a\neq b\neq c\neq a)$.

Definition 5
We say that an ordered triplex $(a, b, c)$ is in a triangular situa$(a,b,c)$tion if
have the following properties:$b*1c=a$$c*2a=b$$a*3b=c$(18)
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Strictly speaking,correspond $to*1,$ correspond $to*2$ and correspond to. Therefore, if necessary, we must say that $(a, b, c)$ is in a $*1*2*3$ -triangular
$b$

$a$

$*3$

$c$

situation.
Definition 6
Letconstructive laws if

$T$
$T$

be a triple-semilattice. We say thathas the following properties:$T$

has the triangle
(21)
(22)
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$(d*1e)*3(d*2e)=d*3e$$(d*1e)*2(d*3e)=d*2e$$(d*3e)*1(d*2e)=d*1e$
for all$d,$ $e\inT.$

Suppose that a triple-semilattice satisfies the identities $((21)\sim(23))$ for $e\in$. Let $a=d*1e,$ $b=d*2e,$ $c=d*3e$ . If $a\neq b\neq c\neq a$, then $(a, b, c)$ is in atriangular situation. That is why we named the identities “the triangle constructivelaws”.
$T$

$d,$

$T$

Definition 7
Let
be a triple-semilattice. We say thatnatural laws if has the following properties:
$T$

$T$

has the triangle$T$

if $x*1y=z$ and $x*2Z=y$ , then $y*3z=x$
if $x*3y=z$ and $x*2Z=y$ , then $y*1z=x$
if $x*1y=z$ and $x*3Z=y$ , then $y*2z=x$

for all$x,$ $y,$ $z\in

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T.$

Suppose $(T, *1, *2, *3)$ and $(T’, *1, *2, *3)$ are triple-semilattices with
Theorem 1
triangle constructive and triangle natural laws. For a mapping : $Tarrow T’,$$\alpha$

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$\{*2, *\}\gg*$$\{*3, *\}\gg*$$\{*1, *\}\gg*$-homomorphism.
-homomorphism $and*2$ -homomorphism, thenThat is, if
Let be a trice (triple-semilattice with roundabout-absorption laws). If has thetriangle constructive laws, then $T$ has the triangle natural laws. Hence, in a certain
sense, trice with triangle constructive laws corresponds to chain (totally ordered set)of lattice theory.
$\alpha is*1$$T$$\alpha is*3$$T$

I am grateful to Professor Shigeki Matsumoto at Konan University for useful
discussions.

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References[1] S. Burris, H.P.Sankappanavar. A Course in Universal Algebra GTM 78 Springer-Verlag, 1981.

[2] G. Birkhoff. Lattice Theory (third ed.) Amer. Math. Soc. Colloq. Publ., 1967.

[3] G. Gr\"atzer. General Lattice Theory (Second ed.) Birkh\"auser, 1998.

[4] K. Horiuchi. Trice and Two delegates operation. Scientiae Mathematicae, 2-3(1999)373-384.
[5] K. Horiuchi, A. Tepav\v{c}evi\v{c}. On distributive trices. Discussiones Mathematicae
General Algebra and Applications, $21(2001)21-29.$[6] K. Horiuchi. Some weak laws on bisemilattice and triple-semilattice. Scientiae
Mathematicae, 59, No.1 (2004): 41-61.