数理解析研究所講究録1093 巻 1999 年 87-92

87

On Gigantic Pairs of Minimal Clones
fffffl$\overline{\pi}$(Machida,$*\mathrm{H}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{S}\mathrm{u}\mathrm{b}\mathrm{a}\mathrm{S}\mathrm{h}\mathrm{i}$$\mathrm{H}\mathrm{a}\mathrm{i}\mathrm{i}\mathrm{m}\mathrm{e}$University,)$*\mathrm{a}\mathrm{n}\mathrm{d}$Ivo G.

$\mathrm{R}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{e}\Gamma \mathrm{g}\dagger$\dag er Universit\’e de Montr\’eal

Introduction and Preliminary Observations

1

In this paper we deal with some problem from the ”clone theory”, which stands as a raeearch area inmultiplevalued logic and universal algebra. Roughly speaking, a clone is a set of operations (funcan introduction to our problem, one may describe thetions) which is closed under composition.
motivation of the problem in very simple and elementary way as follows:
Is it possible to generahize the set of operations {NOT, OR} on the base set $\{0,1\}$ to a set ofoperations on the base set $\{0,1, \ldots,k-1\}(k\geq 2)\nabla$ ”
This is certainly simple, but is too simple and almost nothing is conveyed from this statement. In
order to state the problem more clearly and definitely, we extract the following properties from the set$\uparrow 1\mathrm{A}\mathrm{s}$$n${NOT, OR}.

{NOT} generates a minimal clone .

(i)$\dagger${OR} generates a minimal clone.
(iii) {NOT, OR} is complete \dag er ( $i.e.$ , generates all the operations).
(ii)

Our problem asks if these properties can be generalized into -valued case.
Problem: Does there exist a pair of operations $f,g$ having the above three properties (i), (\"u) and(i\"u), in place of NOT and OR, for arbitrary $k\geq 2$ ?
We shall call this problm Szab\’o’s problem named after the first person who took up the problem.
$k$

Now, we shall fix $k\geq 2$ and set $k=\{0,1, \ldots,k-1\}$ . Let two operations $f(x_{1}, \ldots,X_{m}),g(x_{1}, \ldots,x_{n})$on satisp the three properties (i), (ii) and (i\"u), that is,$k$(i) $[f]$ is a minimal clone.

(ii) $[g]$ is a minimal clone.
(iii) $[f,g]=O_{k}$ ($=\mathrm{t}\mathrm{h}\mathrm{e}$set of all operations on ) .
$k$$\dagger$

Here, for a set of operations,denotes the set of all operations which are generated by $S$ .
Moreover, we assume that both $f$ and depend on every variable, $ie$ . $f$ is an essetially m-variable$S$$[S]^{\mathrm{f}}$$g$operation and is an essentially n–variable operation.
A pair of operations $(f,g)$ which satisfies the above conditions (i), (\"u) and (iii) shall be called agigantic pair of mininal operations.
In the following, some introductory observations are given on the conditions for a pair $(f,g)$ to bea gigantic pair.
$g$$\mathrm{f}$Claim 1 : If $f(a, \ldots , a)=a$ and $g(a, \ldots,a)=a$ hold for some$a$$\in \mathrm{k},$$[f,g]$ is not complete.

This is immediate from the fact that $f(a, \ldots,a)=a$ and $g(a, \ldots,a)=a$ imply $h(a, \ldots,a)=a$ forwhich is obtained by compositions of $f$ and .

any operation

$h$

$g$

means
$\overline{1}$The$8\mathrm{y}\mathrm{m}\mathrm{b}\mathrm{o}\mathrm{l}\dagger$thata$\mathrm{p}\mathrm{r}\mathrm{a}\mathrm{e}\mathrm{i}\mathrm{a}\mathrm{e}$definition for the term is given in Section 2.

88

Clain 2 : If $m\geq 2$ and $[f]$ is a minimal clone, $f(x, \ldots,x)=x$ for every $x\in k$ . (Such operation
is called idempotent ) The same is true for , of course.

$f$$\mathrm{f}.$$g$This is because if $f(a, \ldots,a)=b(a\neq b)$ for some $a,b,$ $[h]\subset[f]$ holds for unary operation
by $h(x)=f(X, \ldots , x)$ and $[f]$ cannot be a minimal clone.

$h$definedAccording to Claims 1 and 2, at least one operation or must be unary operation. On theother hand, the condition (i\"u) clearly requires that at least one operation must not be unary. These
considerations lead to the following.
$f$

Claim 3 : Let $m\leq n$ . Then
2 or more variables $(n\geq 2)$ .

$f$

$g$

is a unary operation $(m=1)$ and$g$is an idempotent operation with

The following fact characterizing a unary operation which generates a minimal clone is well-known.
and easy to$\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{i}\mathfrak{h}\Gamma$

Claim 4 : Suppose a unary operation is not the identity operation. $[f]$ is a minimal clone if andonly if the following (i) or (\"u) holds.
(1) is not surjective and the restriction of to Image $(f)$ is the identity operation on Image $(f)$ .
(2) is a permutation of prime order.
$f$

$f$

$f$

$f$

As we have seen that is an idempotent operation, must be an operation of type (2) in Claim 4
in order to make the condition
hold. Moreover, one can assert, w.l.o.g., that
$f$

$g$

$(‘\dot{\mathrm{u}}\mathrm{i})$$f=(01\ldots p-1)(pp+1\ldots 2p-1)\ldots((\ell-1)p\ldots\ell p-1)$

where $k=\ell p$.
Due to the similar argument with Claim 1 we have:

Claim 5 : If $f(S)\subseteq S$ and $g(S, \ldots,S)\subseteq S$ for some proper subset
Thus, when $f$ is the operation given above and $\ell>1,$$g$$S\subset k,$$[f,g]$ is not complete.

must be an operation which$g(\{0,1, \ldots,p-1\}, \ldots, \{0,1, \ldots,p-1\})\not\subset\{0,1, \ldots,p-1\}$$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathfrak{g}\Gamma,$$e.g.$,.

So far, what we have discussed is merely the very basic properties that operations and must
. These observations are the first step of our study. By extending the considerations nore deeply,
we have obtained the following results which will be stated in detail in Section 3.
(1) We obtained a necessary and sufficient condition for a pair of operations $(f,g)$ to be a gigantic
$f$

$g$

$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{S}}\Phi$pair. (Theorem 3.1)
(2) We verified that a gigantic pair really exists for every $k\neq 2^{t}(t>1)$ by constructing such pairs.
(Theorem 3.3) In our construction is the join operator corraepondin to some semilattice . (Fig. 1)
The reader is invited tothat the premiss in Claim 5 does not hold for this operation and theabove operation .
In the following, due to the lack of space, we shall give only definitions and theorems, but (almost)no proofs.
$\mathrm{f}$$g$$\mathrm{g}$$\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{i}\Psi$

$g$

$f$

2

Definitions

Let
be the set of all n-axy operations from
into
$pr_{i}^{n}(1\leq
i\leq
n)$
the set of all projections
over where$\mathcal{O}_{k}^{(n)}$$k^{n}$$k$$k$

and let. Let beis defined as $pr_{i}^{n}(x1, \ldots,xi, \ldots,xn)=x_{i}$$pr^{n}.\cdot$$o_{k}(=O_{A})= \bigcup_{n=1}^{\infty}o^{(n}k)$$J_{k}$

89

for arbitrary.
in
An operation $f\in O^{(n)}$ is idempotent if $f(x, \ldots,x)\approx x$ . (Here and in the sequel,
identity over $k,$ $i.e.,$ $f(x, \ldots,x)\approx x$ means $f(x, \ldots,x)=x$ for all $x\in k.$ )$(x_{1}, \ldots,x_{n})$Deflnition 2. 1 A subset$k^{n}$$C$(functional) composition.

of$O_{k}$

is$a$clone onif (i)$k$$C$

contains

Definition 2. 2 For a subset of , the done generated bycontaining S. It is denoted by $[S].$ A set is complete if
be obtained by composing operations in.
$S$$\mathcal{O}_{k}$$S$$J_{k}$$\approx$

denotes anand (ii) $C$ is closed underis defined to be the smallest done
, i.e., every operation in
can

$S$

$[S]=\mathcal{O}_{k}$$\mathcal{O}_{k}$$S\cup J_{k}$Definition 2. 3 A clone $C$ on$C’=C$ for any done $C’$ on .

$k$

is

$a$

minimal clone if (i) $C\neq J_{k}$ and (\"u)$J_{k}\subset C’\subseteq C$

implies$k$

Definition 2. 4 An operation on is minimal if (i) it generates a minimal clone and (\"u) every
operation from $[f]$ whose arity is smaller than the anty of is a projection.
$k$

$f$

$f$

Theorem 2. 1 [$Ro\mathit{8}\emptyset$

Every minimal opevation belongs to one of the follouring five types:

1) Unary operations $f$ ( , selfinaps of ) such that either (i) $f^{2}(=f\circ f)=f$ or (ii) $f$ is apemutation of prime order $p(1.\mathrm{e}., f^{p}=id)$ .
2) Idempotent binary opemtions; i.e., $f\in O^{(2)}$ such that $f(x,x)\approx x$ .
3) Majority operations; i.e.,such that $f(x,x,y)\approx f(x,y,x)\approx f(y,x,x)\approx x$ .
4) Semiprojections (or quasiprvjections); i.e., $f\in \mathcal{O}^{\langle n)}(3\leq n\leq k)$ such that there enists $i(1\leq$$k$$1.\mathrm{e}.$$f\in \mathcal{O}^{\langle 3)}$$\ldots,a_{n}\in k$ are not pairurise distinct.
whenever5) If $k=2^{m}$ , the temary operations $f(x,y, z)\approx x+y+z$ where $<k;+>\dot{\epsilon}s$ an $elementa\eta$ 2-gmup(1. , the additive grvup of an $m$ -dimensional vector space over $GF(2)$).

$i\leq n)\mathit{8}atishingf(a1, \ldots , a_{n})=a_{i}$$a_{1},$$\mathrm{e}.$Definition 2. 5 A pair $(f,g)$ of minimd operations is called gigantic if $\{f,g\}$ is complete.

$(i.e.$,$[f,g]=\mathcal{O}.)$

The$\dot{\mathrm{f}}\mathrm{o}$llowing terminology is from universal algebra.

Definition 2. 6 A $=<A;F>is$ an algebra if is a set and $F$ is a set of finitary operations
defined over and taking values in A. When $F=\{f_{1}, \ldots, f_{t}\}$ , A is also expressed $as<A;f_{1},$ $\ldots,f_{t}>$ .
For an dgebra A $=<A;F>$ , a subset $B$ of A
subuniverse of A if $B$ is closed under $eve\eta$$B$
opemtion in F. A subuniverse $of<A;F>is$ proper subuniverse if.
For an algebra A $=<A;F>,$ is congruence of A if is an equivdence relation on
and satisfies the property that for $eve\eta$ operation in $F$ , if is $n- a\eta,$
implies$f(x_{1,\ldots,n}X)\theta f(y_{1}, \ldots , y_{n})$for all $X_{1,\ldots,y_{1},\ldots,yn}X_{n},\in A.$ A congruence 9 of A proper if is
not aequivalence relation. An algebra A simple if it has no prvper congruences.
Moreover, for an algebra A $=<A;F>,$is an automorphism of A if is a permutation
on and satisfies the property that for every operation in $F$ , if is n-ary,, $x_{n}\in A.$ The set of all automorphisms of A is denoted by Aut A. $An$
for all
audomorphism is proper if is not an identity pemutation
of .
$A$

$i\mathit{8}$

$f$

$a$

$\phi\subset B\subset A$$a$$\theta$$A$$\theta$

$a$

$f$

$f$

$x_{1}\theta y_{1},$$\ldots,xn\theta yn$$is$$\theta$$is$$t\gamma\dot{\eta}\dot{m}al$$\varphi$$\varphi$

$A$

$f$

$\varphi(f(x_{1}, \ldots,x_{n}))$$f(\varphi(x_{1}), \ldots,\varphi(X_{n}))=$$f$$x_{1},$ $\ldots$$id_{A}$$\varphi$$\varphi$$A$

FinaUy in this subsection, we supply some definitions conceming relations.

Definition 2. 7 For a set , an -ary relation on is a subset of the Cartesian productn-ary operation in
and an $h- a\eta$ relation on $A,$ is said to preserve if$(f(x_{11,12}x, \ldots,X_{1n}), \ldots,f(xh1,xh2, \ldots,xhn))\in\rho$ .
every$j=1,2,$
implies
for
$A$

$A$

$h$

$O_{A}$$f$$\rho$$f$$\rho$$A^{h}$. For an$(x_{1\mathrm{j}’ j,\ldots,j}x_{2}xh)\in\rho$$\ldots,n$Deflnition 2. 8 Let $k=h^{m}$ where
and $m\geq 1.$ A setof $e_{\Psi}4ivdence$ relations
on is an -regular system if (i) each block (equivalence dass) of every has elements and (\"u)
if B. a block of for all $i=1,$ $\ldots,m$ then $|B_{1}\cap\cdots\cap B_{m}|\geq 1$. Nezt, the relation detemined byis the relationsuch that for each $i=1,$ $\ldots,m$ it hol&defined as the set of all
$k$

$h$

$\dot{u}$

$T$

that

$h>\mathit{2}$.

$\theta_{i}$$\theta$$(a_{1}, \ldots,a_{h})\in k^{h}$$\lambda_{T}$$a,.\theta_{i}a_{s}$$T=\{\theta_{1}, \ldots,\theta_{m}\}$for some$1\leq r<s\leq h$

.

$h$

90

Main Results

3

denote by
the set of all pemutations of unth $\ell:=k/p$ cydes
Definition 3. 1 For a$f\in F_{p}$ have cydes.
.
LetlengthofAn equivdence relation on is transversal to if there nist 1) an equivalence relation on
distinct fivm the least equivdence relation on and 2) an element
such thatfor each$di_{\dot{\mathrm{W}}Sor}pofk$$p$$\theta$$k$$F_{\mathrm{p}}$$C_{0},$$C_{t-1}$$\ldots,$

$k$

$f$

$\lambda$$\theta=\{(f^{m}(c_{\mathrm{t}}), fm(C_{j}))|i\lambda j, 0\leq m\leq p-1\}$.

A pemutation $\psi ofk$ is orthogonal to if 1)
for some primeand 3) if $q\neq p$ then each cyde of meets every Ci in at most a singleton.
$f$$di_{\dot{\mathfrak{R}}SO}fqofk,$ $2$$\psi\in F_{q}$$\ell$$i\in\ell$$c_{i}\in C_{i}$$\ell$) $f\circ\psi=\psi\circ f$$\psi$

Now, we characterize gigantic pairs.
Theorem 3. 1 Let $f\in O^{(m)}$ and $g\in O^{\{n)}$ where $m\leq n$ and let $\mathrm{A}:=<k;g>$ . Then the pair $(f,g)$gigantic if and only if$i\mathit{8}$(i) $m=1$ and(ii) $n>1$ and(i\"u)$f\in F_{\mathrm{p}}$$g$

for some prime divisor of
$p$

$k$

(urith cycles$c_{0},$ $\ldots,c_{\ell_{-}1}$),

is minimd,$C_{1}.\cup\cdots\cup c_{:_{h}}$is not a prvper subuniverse of A for any $0\leq i_{1}<\cdots<i_{h}\leq\ell-1$ ,(iv) No congruence of A is tmnsversal to ,$f$(v) No automorphism of A is orthogond to ,$f$(vi) Let $k=h^{m}$ where $h>2$ and $m>1$ . If there nist. a) a pemutation of $m$ of order 1 or$\ldots,f_{m}-1$
permutationsof such thatfor each cydeof ,permutation
every
point
is of orvier 1 or andof the
for
fixed
$xed$
permutation
point of theis $fi_{Xed- p}oint- fiu$ and offor someoder
and b) a bijection$\cdot$$\varphi$$\alpha)f_{d_{0}}fd1\ldots fd_{\mathrm{p}-1}=id_{h}$$(d_{0}\ldots d_{\mathrm{p}-1})$$i$$\beta)$$\gamma)$

$p$

$p$

and

$h$

$f_{0},$

fi.

$j$$\varphi$$f_{i}$$\varphi$$p$$f_{\mathrm{j}}$$\varphi$$\psi:x\mapsto x=(\wedge x^{\mathrm{t}0)\ldots,\mathrm{t})},Xm-1)$

of

$k$

onto

$h^{m}$

such that for $dlx\in k$$fm-1(X)\mathrm{t}\varphi(m-1)))$$\overline{f(X})=(f_{0}(x)\mathrm{t}\varphi(0)),$$\ldots,$

then

$g$

,

does not preserve the h-ary relation{$(a_{1},$$\ldots,a_{h})\in k^{h}|\#\{a_{1}^{\mathrm{t}},.,ai)..(p*)\}<p-1$(vii) a) Let $k=p^{m},$ ) let $x\mapsto x\wedge be$ a bijection fiom
$dlm\cross 1$ matrices over $p$ ),) let for $dlx\in k$$\mathrm{b}$

$k$

onto

for every $i=0,$ $\ldots,m-1$ }.
$p^{m}$(the latter considered as the set of$\mathrm{c}$$\overline{f(x}\rangle=A^{\wedge}X+B$

where $A=P^{-1}JP,$ $B=P^{-1}B’$ with $P$ a nonsingular (over $GF(p)$)
the $m\cross mJo7danmat\dot{m}$ urith diagon.al $(1, \ldots, 1)$ and blocks of sizes$(b_{1}, \ldots,b_{m})\in p^{m}$ isthat$m\cross mmat\gamma\dot{\tau}x$$J$$\ell_{1},$$\ldots,\ell_{h}$$\mathit{8}uch$$(I+J+J^{2\mathrm{p}}+\cdots+J-1)B’=o_{m}\mathrm{x}1$and$b_{\ell_{1}+\cdots+\ell_{u}}\neq 0$

for some$1\leq u\leq h$,. Then does not preserve the quaternary relation$g$$\{(_{X},y,z,t)\in k^{4}|t=x\ominus y\oplus z\wedge\}\wedge\wedge\wedge$.

over ,$p$

and $B’=$91

The proof of Theorem 3.1 heavily relies on the fouowing theorem.
] For a set $B$ of surjective operations containing an essentidly more than
Theorem 3. 2 [Rounary operation, the set
complete if and oily if$(A)<k;B>is$ simple and has no prvper subuniverse and no prvper automorphism,$(B)$
If $k=p^{m}$ for a prime and $m\geq 1$ and $if<k;+>is$ an elementaw abelian -group then some$b\in B$ does not preserve the quaternary relation$70\mathrm{B}$$Bi\mathit{8}$

$p$

$p$

$\{(x,y,Z,x-y+z)|x,y,z\in k\}$ ,

and
$(C)$

If $k=h^{m}$ urith $h>2$ and $m>1$ and.

preserve$T$

is an -regudar system on
$h$

$k$

then some $b\in B$ does not$\lambda_{T}$Example. Let $k=2$ (Boolean case). There are 4 nonunary minimal operations, namely,and, where V and are OR and AND, $d(x,y,z)\approx$ ( A ) ( A ) $\vee(z\wedge x)$ and $r(x,y, z)\approx x\oplus y\oplus z$ .
With the unary non-identity operation (NOT), two pairs
andare shown to be gigantic.
These are the only gigantic pairs for $k=2$ .
$\vee,$$r$$\wedge$

$x$

$y$

${ }$

$(\neg,\vee)$$\neg$

Next, we state that a gigantic pair really exists for every$\wedge,$

$d$

$z$

$y$

$k$

$(\neg,\wedge)$

which is not a power of 2.

Theorem 3. 3 For $eve\eta k$ where $k\neq 2^{t}(t>1)$ , there enists a gigantic pair.
Proof If is a prime, the claim can be verified for thetwo operations: Define $f(x):\approx x$ el
and $g(x,y)$ such that $g(x,y)=x$ if $x=y$ and $g(x,y)=k-1$ if $x\neq y$ for all $x,y\in k$ . It is easy to see$k-\mathit{2}\leq k-1$ .
that is minimal; in fact, is ajoin whose order is $0\leq k-1,$
Suppose that is a composite number. Let denote the greatest prime divisor of and let $\ell:=k/p$
where $\ell>1$ .
The following unary operation and binary operation on suffice for our purpose.
We define $f$ by setting$k$$\mathrm{f}\mathrm{o}\mathrm{u}_{0}\mathrm{W}\mathrm{i}\mathrm{n}\mathrm{g}$

$g$

$g$

$\mathrm{s}\mathfrak{W}\mathrm{l}\mathrm{i}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}$$\ldots,$

$k$

$k$

$p$

$f$

$ip+j+1$$f(ip+j):=\{$ $ip$
Thus,

$f$

$k$

$g$

$i\in\ell,$$i\in\ell,$$0\leq j<p-1$ ,$j=p-1$ .

is expressed in the cyclic notation as$f=(01\cdots p-1)(pp+1\cdots 2p-1)\cdots((\ell-1)p(\ell-1)p+1\cdots k-1)$

An algebra
idempotent ($\mathrm{A}=<k;>\mathrm{i}\mathrm{s}$$\dot{u}e.$, it satisfies

a semilattice if the binary operator$x(y\vee z)=(x\vee y)\vee Z,$$x\vee y=y\vee X$

${ }$

and

.

is associative, commutative and$x\vee x=x$

for all $x,y\in k$ ). The binary relation
corresponding to A is defined by setting $a\leq b$ whenever$\vee b=b$ . It is well-known thata partial order in which $a\vee b$ is the join of and . Conversely,an orderin which each pair has ajoin determines a semilattice.
Let be the following semilattice operator V on . For every $x,$ in set$\leq \mathrm{o}\mathrm{n}k$

$a$

$b$

$a$

$\leq \mathrm{i}\mathrm{s}$$\leq \mathrm{o}\mathrm{n}k$

$k$

$g$

$x\vee y=\{$

$y$

$x$

$x=y$ ,$(i+1)p$$x=ip+j,$ $y=ip+j’$$(0\leq i\leq$ $\ell-2,1\leq$0$.
otherwise.

$k$j,$ $j’\leq p-1,$$j\neq j’\rangle$,

in Fig. 1. Notice that it
The Hasse diagram of the order corresponding to the semilattice
is a tree.
The pair $(f, )$ is proved to satisfy the conditions (i)-(v\"u) in Theorem 3.1 and so it is gigantic.
$<k;>\mathrm{i}\mathrm{s}$$\square$92

Figure 1: The semilattice for the operation V

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