ELEMENTARY PROPERTIES OF SEMIGROUPS

OF TRANSFORMATIONS OF ORDERED SETS

Y u . M. V a z h e n i n UDC 51 01+518.5

I_~t g ~ be some model, and S ( ~ ) a semigroup (somet imes a par t ia l groupoid) of cer ta in t r a n s - format ions of this model. In studying semigroups of t r ans fo rmat ions as abs t rac t semigroups , one was pr incipal ly in te res ted in those p roper t i e s which a re re ta ined under i somorph i sms . Examples are the s ea r c h for the abs t rac t cha r ac t e r i s t i c s of S(~gZ) (see, for example , [1-4]), a descr ip t ion of all its con- gruences (see, for example, [1, 5]), the explanation of the definability of model ~ by semigroup 5 ( ~ ) : of a quas io rde red set by the semigroup of all isotone t rans format ions [6], of an o rde red set by the semi - group of all d i rec ted t rans format ions [7], of an a r b i t r a r y graph by the part ial groupoid of all d i rec ted t r ans fo rmat ions [8], etc. It was L. N. Shevrin who posed the task of studying the e lementa ry p roper t i es of semigroups of t r ans fo rma t ions . What, for example, is it possible to say about the monotypical models and ~ whose semigroups 5 ( ~ ) and S ( ~ ) a re e lementa r i ly equivalent? Will the semigroups S ( ~ ) and S ( ~ ) be e lementa r i ly equivalent if models ~ and ~Z are e lementa r i ly equivalent? These and re la ted questions natural ly a r i se in investigating the capabil i t ies of the language of the r e s t r i c t e d p r e - dicate calculus for the study of semigroups of t rm~sformations.

In the presen t paper we shall adduce some resu l t s in the aforementioned di rec t ions . We shall study the case when ~ is an o r d e r e d (in the sense of par t ia l ly ordered) set , while S(ff~Z) is the s e mi g ro u p of all isotone transformations (§ I), the semigroup of all inf-endomorphisms (§ 2), and the semigroup of all directed transformations (§3). For the types of sen~groups listed we shall study the questions posed above. Tile basic results are contained in Theorems 1.1, 2.1, 3.1, and 3.2.

In what follows we shall use the symbols ~< and ~ to denote ordering relationships on given sets, and the symbols >i and ~ to denote the corresponding inverse relationships, where, as usual, the notation ~ < ~ (~ <3 ~ ) , d e n o t e s ~ . < ~ ( ~ ) and ~ ¢ ~ .

§ 1 . S e m i g r o u p s o f A l l I s o t o n e T r a n s f o r m a t i o n s

Let ( f 2 , - . < ) be an o rde r ed (o.) set . By _T(hr2fi) we denote the semigroup of all its isotone t r ans - format ions , i .e . , t r ans format ions ~c for which it follows f rom &..< ~ that ~c~ ~ Jc ~ for any £ , ] c J~ .

In this section we shall prove

THEOREM 1.1. Let (~(2, ~< ) and (f)..,~ ~ ) be o. se ts . If the semigroups 7"(~.,.< ) and ~ ( ~ ) are e l emen ta r i ly equivalent, then (o'2, ..<) is e l ementa r i ly equivalent to one of the sets ( ~ , t ~ ) , ( ~ ~>~). The conver se is untrue.

Before proceeding to the proof, we make the following r e m a r k . Since the relat ionship & is ref lexive, the set S¢ (~'~) of all t rmlsformat ions ~ £ , for which o ~ 5r~ = ~ , is a subset of the semigroup 7 (~(2, ~ )

With this, for any ~c e ] - (~r~, ~ ) the equation ~ g ~c =o _ ~ holds, while the equation Jco~ = Q 2 holds

if and only if m ~ = ~ .

If _(2 is a s ing le-e lement set the semigroup -~ (b'-2~, -.< ) also has a single e lement . F r o m the e l e - menta ry equivalence of . /(J-2, ..< ) and _7"(ff2, ~ &) it follows that f f ( ~ , ~ ) also has a single e lement . This means that ~ t has a single e lement and ( J ~ , ~ ) is e l emen ta r i ly equivalent to (~ r -~ / ~ ) .

T rans la t ed f rom Algebra i Logika, Vol. 9, No. 3, pp. 281-301, May-June, 1970. Original ar t ic le sub- mit ted October 28, 1969.

© 1971 Consultants Bureau, a division o[ Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced [or any purpose whatsoever without permission of the publisher. A copy o[ this article is available [rom the publisher [or $15.00.

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For the r e m a i n d e r of § 1 we shal l a s s u m e that g'2 is not a s i ng l e - e l emen t se t .

We shall need the following

LEMMA 1.1. If oc__</3, 3 ~ (~-,p, ~, 3 ~ C2) we can then find a t r a n s f o r m a t i o n . ~ Z (~'-2, ~ ) such that ac ~ =- oc, and ~¢~ -~/~.

Fo r the proof of the l e m m a it suff ices to cons ider the t r a n s f o r m a t i o n ~ f ~ ~ , given by the conditions

We speci fy the p red ica t e s H,/7 , and C by the following fo rmulas :

H C~:) ~--~y)(-=y = =:~, (I)

(~, y ; g,U ).~-~z}(H (x)&/-/(~/)~ l-/ ($)&./-/(U)&z~g=x&(zu:y), (~.) 17

O(~,~f) ¢/~lCulCHcx)~ H C y )~ /~ / mC~C=~&nC~,u: x,y)-,-Fl(x,y ; z,u))). (3)

If ~ z ~ , then ~=u----- ~c for any isotone t r a n s f o r m a t i o n of the o. se t ( ~ , - ' ~ ) . If acjy-~ag for any _ueZ

(~...~) , then, by takir.~, t r an s fo rm a t i on ~ as ~/ , we obtain :r.a ~=ac~-=.v.~ for any _~ ~ ~ , i . e . , ~ = c z x g .

Consequently, the fo rm u l a /-/r-r.~ is t rue on semigroup _2"(~"~, ~ ) if and only if z ~ - . ~ , ( ~ ) . The t ru th of fo rmula F l ( a c , y i z , u) on semigroup .T ( ~ , ~ ) means that x , j / ,~ ,~ze . f f , C~'-~) and czg= ac , c z u = y for some ~z~3" (. .~, ,~ ) or , in o ther words ,

The following l e m m a explains the meaning of p red ica te O on s emig roup .T C ~ , K= ) where ~.~ is an o rde r re la t ionship different f r o m equal i ty . In case ~ is the equal i ty re la t ionship , C C ~ , c., 3) is t rue on T C ~ , ~ < ~ ) ff and only if o(~=j3 .

LEMMA 1.2. F o r fo rmula Ccc~,a#) to be t rue on 2 " ( ~ ' ~ , ~ ) , it is n e c e s s a r y and suff icient that one of the re la t ionships o¢-c/~ or ~ < oc hold.

Proof . Let the fo rmula C(a . , , ct~ ~ be t rue and let , for e x a m p l e , / 3 ~ o c . Since the re la t ionsh ip is not the equali ty, we can find e l emen t s ~ , ~ for which ~-" 8 , and whence, by L e m m a 1.1, the t ruth of the fo rmula /'7 (czj ,~$, a~,, a/3 ) , and, consequently, the t ruth of the fo rmu la /7 (~z~,~/s ; c~/l , aS ) • Conse-

quently, c z c z ~ o ~ , and~czB=~ ~ , , i . e . , ~z]---- o¢ and c ~ = / ~ , for some cz~Z($'~, ,~), t~,yvirtue o f t h e

isotonici ty of ~ i t follows f r o m the two las t equations that ~- - /~ . Analogously, in the case ¢~,~/~ we obtain the re la t ionship /3<o~ .

Converse ly , let oc-~/~ and let the fo rmula f 7 ( ~ z ~ ; ~zo~,~#) be t rue on the semigroup _Z'($~,--- ~ )

for some different ~, ~": ~ . Then, by the definition of p red ica te r7 [see (2)] we have the re la t ionsh ip ~ ~. According to L e m m a 1.1 the re ex i s t s ~-Z(:~4~, ~ ) (and, speci f ica l ly : ~=~o¢#~ ) for which the

following equations hold: ff]=o~,##--~ i .e . , ~ z _~o~ , ~ z ~ - - ~/~, and, consequently, t l i d t ru th of the fo rmula

r7 ~ , ~ ; ~ , , ~ .

The case /3 --oc is handled analogously. The l e m m a is p roven .

We speci fy the p red ica te P by the fo rmula

p¢__c.,y; z, ~) ~'/(n~,y; z, u)~ n ~, .=; z, ,-,)) v C..~ =~,~ H~..r..~). (4)

In what follows we shall denote by the symbol P ( , ; n, ~) the b inary p red ica te defined by the p red ica te (~c D', ~, ~) for fixed a , and z? .

Let OZ (.v~, ...,~=~; ~) be an ax iom (closed formula) of the s ignature , containing the unique e x t r a - logical two-p lace pred ica te symbol ~- . If this ax iom is t rue (false) on the o. set ( ~ , ~ ) where the role of w is placed by ~ , we adopt the convention of denoting this fact by "ax iom OL (x~ . . . . . ~cn ; ..< ) is t rue (false)" ; to be sure , ins tead of the re la t ionship ~ we could cons ider the re la t ionsh ips ~ , ,~ , L~ . We shal l speak analogously of the t ru th (falsehood) of the ax iom O'Lca% .... ,~n ; p ( , ; a , ~)) unders tanding by this the t ru th (falsehood) of the axiom OL(.z~ .... . :c n ;?t) on the given semigroup , where the ro le of ~ is played by P ( , ; a , d ) .

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L~.MMA 1.3. Le t ( ~ ( ~ ) be an a r b i t r a r y o. se t . If the ax iom

£* C~ ..... ~ ;..< ) d~---Z ( ~,~c,),,. (~,~c~) ~ (~,..., :c~; ~ ) ,

is t r ue , w h e r e ~ V , - z and ¢P conta ins no quan t i f i e r s , then the fol lowing ax iom is t r u e (where ~ is obta ined f r o m ~r by r e l a t i v i z a t i on with r e s p e c t to /4 [10])

s~C-~,,...~,~; ~{., ;,~,oz)) ~ a ) ~ ) ( c ¢ ~ , ~ H (~:,,...,~; pc, ; ¢z, ~)).

Conve r se ly , if the ax iom ~ $ (-~ ..... x ~ ; ~ (, ;c~, ~)) is t rue , then one of the fol lowing ax ioms is t rue £, (~1 .... , '2 :n;~) , ~ (Jcl ..... ~,v ; ~ ) -

Proof. Since isomorphic models are e lementar i ly equivalent, i t suffices to establ ish the following: one can find or, ~e.~CQ) such that the re lat ionship D ( , ; ~, ~) w i l l be an order relat ionship, and the o r - dered sets (,_~.4) and ( ~ ($"2); ~C, ; c¢, ~)) are isomorphic (with th is , the lemma~s second assert ion is easi ly obtained, star t ing f rom the def ini t ion of predicate # ).

The correspondence ~ by which each element ~e3~ has made correspond to i t an elemento_¢e S~(3~). is , obviously, a one-to-one mapping of a'~ onto St(d2). By the very def ini t ion of predicate z) [see (4)], the fornmla P(cz~ ,o$ ;c~ ,~/s ), where oc #=/3 i f ~ is the equali ty relat ionship, and oc.</~-otherwise, w i l l hold i f and only i f ~ Consequently, ~o is an isomorphism of ¢ -~ ,~) and (-~I (g~); p [ , ;o~,~p)). The lemma is proven.

We now turn immediately to the proof of Theorem 1.1. Let the semigroups E C Q , ~ ) and _7 (~2~__~) be e lementar i ly equivalent. Applying the f i rs t , then the second, assert ion of Lemma 1.3, we readi ly show tha t if the a x i o m ~5 ( ~ .... x . ; _~) is t rue then one of the ax ioms . , ~ ( x , . . . , ~ n ; , a ) , ::5 (..~;,...,.z=~ ; ~ ) is t r ue . Hence , if kS(.x:,...,.:v,.,,;~) and . ~ (ac,,+4 ..... a S , ; ~ ) a r e a r b i t r a r y t rue ax ioms , then e i the r the

ax ioms ~C~% .... ,a:,z;-..~) and ,tG2(*~,..., ,:v,.,.;.~ ) are t rue , o r the ax ioms ~,¢:c~ . . . . . ar, z ; ~ ~ and ~2 (,.v...,.,+,, . . . . , :c.,;g), s ince o the rwi se the ax iom

will be t r u e and the a x i o m s

" ~ ( % . . . . . "~tr~, .2:,-~+,, . . . . , ~ ; ~ ) and ~ C.-.~ ..... ,..C:,..,.~,'.~m+,,,".,-"crt;~')

will be fa l se , which c on t r a d i c t s L e m m a 1.3. Thus , only two c a s e s a r e poss ib l e : f r o m the t ru th of any a x i o m on the o. se t (..63~_~) fol lows the t ru th of th i s ax iom on the o . se t (~2,~) o r f r o m the t ru th of any a x i o m on the o. se t (_Q, ~ ) fol lows i ts t ru th on the o. s e t ( g : 2 ; ~ ) . In the f irs¢ c a s e the o. se t s (g'2,~_~) and (g~,~ ~ ) a r e e l e m e n t a r i l y equiva len t , in the s econd c a s e the s e t s ( ~ , ~ ) and ( ~ ' 0 , ~ ) .

We c o n s t r u c t an example of two e l e m e n t a r i l y equiva len t o. se t s t,:2 -= ~ and ,g9 . M ~ such that the s e m i g r o u p s ! ( . ~ - ) , ~ ) and E - ( D ~ ) a r e not e l e m e n t a r i l y equiva len t . This will comple te the p roof of T h e o r e m 1.1.

I_~t CQz, ;T, ), ( g~z , ' ~ ) , ( f f2 f , ~ / ) be i s o m o r p h i c l i nea r ly o r d e r e d (1.o.) se t s with the ca rd ina l i t y

of the con t inuum, and let 5 f be an i s o m o r p h i s m of ~£3,, 'z, ) on ~ ) z , "Tz ) ° F u r t h e r m o r e , let (,~2./, ,zzz/)

be a 1.o. se t of countable ca rd ina l i t y which is e l e m e n t a r i l y equiva len t to the 1.o. set , ~'2, ;z, ) . We a s -

s u m e tha t . 4 - ~ , f ~ Q = ~ and g ~ 2 , / ~ f f 2 j = ~ We se t ._~2 g2 ~ ~ , ~ - ~ ; ' ~ , ~ , ? z / a n d , on the se t s . Q andg-2 ~,

we define an o r d e r i n g in the fol lowing way: f o r any __4, y ~ ¢2 ( r e spec t ive ly , g , / ~ - g-~/) we se t _4 ~

(_~-<q ~ ' ) if and only if one of the r e l a t i onsh ips ~ C ¢ , ~ ) , % c.__~, ~) ( r e spec t ive ly , %/C~', ~/), ' z~z / ( -~ ~} )

is t r u e . Since the 1.o. se t s (.-Qg,~[) and ~g-2/, ~ : ) , where ~ = / , ~ , a r e p a i r w i s e e l e m e n t a r i l y equivalent , it is e a s y to c o n s t r u c t a winning s t r a t e g y for p l a y e r II in the game ~a ( ( -Q,~) , (£2 f ~ ' ) ) fo r any ~- , which p r o v e s that the c o n s t r u c t e d o. s e t s a r e e l e m e n t a r y equ iva len t (see [9]). We show, however , t ha t the ax iom

(3c~ (J:) (.~ C~c) - ~ ~2~ ~ ~: ~ ~ ¢< ~ - ~ ) (5)

t r u e on s e m i g r o u p Z ( ~ , ~ ) is fa l se o n s e m i g r o u p r ( ~ , ~ ) . Indeed, ff we choose as c~ the t r a n s f o r m a - t ion of se t ~2 for which ~_~=f.£ when ~ £ ~ and c ~ £ = ~ - / ~ when g ' ~ - J z then, obviously , ~ _ 7 ( ~ 2 , ~

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-/~ :. ' '~-; - - ~ (when _¢ and, f o r a n y ~ c ~ 2 w e h a v e ~ c z ¢ 4 = ~ ¢ , c z c ~ c ~ , Q ¢ = c ~ ¢ and c~czc~ : ~ ~zd--" '_¢

e _42~ or _ d ~ r e spec t ive ly ) . Thus, the ax iom of (5) is t rue on semig roup f (.Q,.--~) Let us now

a s s u m e that this ax iom is a lso t rue on the semig roup __r(g2/~) . F r o m the fact that c z c ~ z=_= ~ , for any

_~ ~ e ~;~ ~ it follows that cz is a one- to -one mapping of g-2 ~ onto if2 / With this , c ~ ~ ~ ~-_A ~ if and

only if e i the r _g r~_Q~/ and ~ ~/~dz, or •'~c ," )~ and ~/,-:: .4~f (otherwise, in view of the i sotonic i ty of ~z

and the p r o p e r t y .~,~d' ~ _d ~ f r o m the re la t ionship _~ ~ ~ ~ follows the re la t ionship ~ _~' and

analogously, f r o m the re la t ionship ~ '--_~_A ~ follows the re la t ionship ~ z_~' ~;~, i .e . , a lways _~ ~ ~ ,

which is a contradict ion) . Thus, ~: is a one- to -one mapping of g'2/ on ff~f and of ..C~z~ on ~ '~/ , but the

se t g ~ / has the card ina l i ty of the continuum while se t $'P~ is countable. This cont radic t ion p roves the

fa ls i ty of ax iom (5) on semigroup 7-(fi2~ ..4 ) . T h e o r e m 1.1 is p roven ,

We r e m a r k that for the proof of L e m m a 1,1, which was used for ve r i fy ing all the subsequent a s s e r - t ions, we used only t r an s fo rm a t i ons the se t s of whose images cons is ted of no more than two e l emen t s . T h e r e f o r e , denoting by _~ (~-2. ~ ) the subsemigroup of s emig roup .Z(._~, ~ ) consis t ing of t r a n s f o r m a -

t i o n s , , for which the card ina l i ty of se t ~*: g-2 does not exceed the ca rd ina l i ty of number ~ , we read i ly ve r i fy the val idi ty of

THEOREM 1.2. Let (_Q, ~ ) and (g-2/,-~_ ~ ) be a r b i t r a r y o. s e t s , If the s emig roups ! / z ( f f f l __~)

and _~ ( i f 2 ' -~ , ~ ) , where / -z 1> Z a re e l emen ta r i l y equivalent , then the o. set ( g ' ~ , ~ ) is e l e m e n t a r i l y

equivalent to one of the se t s (~2( ,~Z), (g-J;~ ' - ) .

It is unknown, however , whether the conver se is val id when / . z is f ini te.

§ 2 . S e m i g r o u p s o f A l l i n f - E n d o m o r p h i s m s

A s a b o v e , let ( g 2 , ~ b e a n o , se t . W e d e n o t e b y K I g P ~ ) the semigroup of all i ts i n f - e n d o m o r - ph i sms , i .e . , t r a n s f o r m a t i o n s -v for which it follows f r o m ¢-----~n~? ~1 that m.=Zrt~.~, .-c~) for any

An exact bound (upper or lower) of e l emen t s c~,/~+ e £2 is cal led nontr iv ia l if ~ and /~ a r e i n c o m - mensurab le (i.e., oc,~fl and ~ o c ) and, o therwise , t r i v i a l .

This sec t ion is devoted to the proof of the following t h e o r e m .

T H E O R E M 2 . 1 . Let ( g 2 , - ~ ) and ( ~ 2 , ~ ) b e o . se t s , a n d l e t t h e s e m i g r o u p s ~ ( . g 2 , ~ ) and ,~t_Q~, ~ ) be e l emen ta r i l y equivalent . Then, a) if al l the exact bounds of se t (_42, , ,~) a r e t r iv i a l then

tgP,.~ ~ is e l emen ta r i l y equivalent to one of the o. se t s (~"-2 ~ ~ ) , t £-2 ( ~ ) ; b) if the re is a nontr iv ia l /

b o u n d i n o , s e t ( g 2 -~) t h e n t h e o. se ts ¢fi2. 4 ) and (g -2 ,~) a r e e l e m e n t a r i l y equivalent .

The conve r se is fa lse: the e l e m e n t a r y equivalence of (g'2, ..4 ) and (A~; ~ does not, in genera l , entail the e l e m e n t a r y equivalence of

It is e a sy to see that

E(X2,.~) and E ( X 2 : 4 ) .

5 7 C,A'3_,)~_ EC.C-Z,~).

This al lows us to use in this sect ion the p r o p e r t i e s of t r a n s f o r m a t i o n a.g (see the r e m a r k a f t e r the f o r m u l a - t-ion of T h e o r e m 1.1). We note, m o r e o v e r , that each i n f - endomorph i sm is an isotone t r a n s f o r m a t i o n .

In the p resen t case , the following l e m m a , analogous to L e m m a 1,1, is val id .

E M M A 2.1. If

we can then find a t r a n s f o r m a t i o n x e g(,.(2, ~) such that

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To prove this l e m m a it suff ices to cons ider the t r a n s f o r m a t i o n c ~ specif ied by the conditions:

c ~ / ~ =oc, i f g ; ~ . and c ~ 3 g = ~ if ~ [4].

Thanks to what has been said, we can eas i ly ve r i fy that L e m m a s 1.2 and 1.3 will be val id for s e m i - group ES.~ .~) if in all t h e r e q u i s i t e p l a c e s the symbol 7"(X~,~) is r ep l aced by ~ ' ( . ~ . ~ ) and if, i n t h e proof of L e m m a 1.2, one u se s L e m m a 2.1 ins tead of L e m m a 1.1. In what follows, when r e f e r e n c e is made to the f i r s t sec t ion , we shal l be a s suming that the n e c e s s a r y r e p l a c e m e n t s have been made.

The following l e m m a shows that the cons idera t ion of case a) of T h e o r e m 2.1 reduces to T h e o r e m 1.1.

LEMMA 2.2. The equation K(._O_,,~) =~C,.Q,. ~ ) holds if and only if all the exact lower bounds in o. se t C _~ , ~ ) a r e t r i v i a l .

P roof . Let o c = Lrt:~ ~/~,~) be a nontr ivia l bound. We cons ider the t r a n s f o r m a t i o n ~ ) ~ 7 ~ ' ~ , ~ ) defined in the proof of L e m m a 1.1. Since

t r a n s f o r m a t i o n ~ does not belong to s emig roup ~CA-2,~<.). Consequently, i f ~(LC2, ~ )= .Z(_~ ,~<) ,

then all the exact lower bounds of ( ~ , ~ ) a r e t r i v i a l .

Conver se ly , le t the re be no nontr iv ia l exac t lower bounds in (~ ;~ ,~ ) , i .e . , ~----5~-f(/~,~) if and only

i f / ~ o r ~-~/5 . F r o m this obviously follows the inclusion _ 7 ( ~ ) ~ ( ~ , ~ ] and s ince the i nve r se

inclusion a lso holds, we have proven what was to be proved.

Before turning to case b), we make s e v e r a l r e m a r k s . Let ( ~ , ~ < ) be an o. se t with nontr ivial exact (upper or lower) bounds or a s ing l e -e l emen t o. se t . Close to our goal will be the cons t ruc t ion of a fo rmula in the f i r s t - o r d e r language with a unique functional symbol for the semigroup operat ion, giving a two-place p red ica te A4 such that the o. se t (ff2,~<) is i somorph ic to se t .Er(g-2) with re la t ion ~ .

We spec i fy th ree p red ica t e s f" . 7- and /7 :

/ " (~ , ~ ,~ ; ~ , v? ~ ( w)(ccu, 0),% p C.~, ~/; ~,v) ,~ p(~,~; u,o)

&CPCu& y; ~, ~))&PC~,z; ~z, v )--*-pc~,w; u , v ))) , (6)

T ( x , y ) '~/(z)(u)CO)(CCx,_~,~F~,~, a i ~ , .u ) - - , - ~ = ~ v ~ = v ) ) , (7)

C ~, y) _d-/oz)Ca)C v~CW)Ccc.v,y),~(r(,~,~,~,yr~f(wgw~,~vs.v,.qj~). (8)

If the f o r m u l a s /'(x,.V,~z;u,o), TC~,.V),-/~C",.V) a re t rue on semigroup E(~;2,~) then the equations ~ - - ~ , ~ , _V=qz, g----o/,u--aB, z ) = a ~ will hold for some ~,fl,~,,~, 9 e ~ [see (3) and (4)], and these f o r -

mulas will have the following r e s p e c t i v e meanings (cf., the r e m a r k a f t e r the formulat ion of Theo rem 1.1); e i ther 3-~ ~) and then¢~ = ~rt~ ( / ~ ) or ~ < 3 , a n d t h e n c ~ - - s ~ p C$,#) ; ~ and /3 a re different c o m - mensu rab l e e l emen t s such that the o. se t [A'2.~) when ~ < ~ (/3 <¢c) does not have nontr ivial exact lower (upper) bounds; ~ and /3 a r e di f ferent c o m m e n s u r a b l e e l ements such that t r a n s f o r m a t i o n s of the s e m i - group ~ ( g ' J , ~-) r e ta in , where ~ < / a ( f ,~ oc), any exact lower (upper) bound. Let us e lucidate , for

example , the las t a s s e r t i o n . Let ¢¢</~ ( , ~ < ~ ) . Then f ' ( , zg ,~8~v ;~z~ ,~ a) means that ]----- ~rt:~(8,?) ( respec t ive ly , ~'=.su.p(:~', ~)) , and the fo rmula f ' c w a ~ , W a ~ , u ~ ; ~ , a /a ) means that u) ~ ~ 5Tt~(w3,W~)

( respec t ive ly , y--s pc a, ) for u ) ~ E ( ~ , i ) .

By means of the p r ed i ca t e s ~ f', T and ~ [see (4), (6)-(8)], we define p red ica te ,w by the fo rmula

where

,:p ~,e C~ca, g ) ~ P c~, y ; ~z, dj& C,.z. c 4 a ) v r ed , a))).

(9)

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It is c l e a r tha t M is a b ina ry r e l a t i o n on ~ , (~'-~) . M o r e o v e r , the fol lowing l e m m a is va l id .

LEMMA 2.3. The o r d e r e d se t (~'~, ~ ) is i s o m o r p h i c to se t "~1 (_C~) with r e l a t i on M .

P r o o f . We show tha t the o n e - t o - o n e mapping of se t - ~ on s e t "-~7 C A ) , which puts into c o r r e s p o n - dence with e a c h e l eme n t ~ e _ ~ the t r a n s f o r m a t i o n ~ j , is an i s o m o r p h i s m of C-4~, ~ ) on (.~,(_C2),M) .

We f i r s t note tha t if ~ = {¢r..~ then "~1 (~"~) = {¢z~ ~1 and, by v i r tue of (9), MCCZ~, r , , ) is t r ue , i .e . , "-~1 ( ~ )

is a s i n g l e - e l e m e n t o. set .

In what fol lows we a s s u m e that t he re is at l eas t one non t r iv ia l exac t bound in (~'~,__~) . We divide the p roof into two c a s e s .

F i r s t c a s e : in the o. se t ($ '~ ,~) t h e r e is a non t r iv ia l exac t upper bound /3 = 5u.p(o¢,~) . We se t

~---cz ¢, ~ = a # . Then, if f o r m u l a / " (~z~ ,aT ,G¢ ;c~ , b ) is t rue then, s ince oc ~23 , we have ~ = ~.~F(~,.~ ) and, f o r any r . J e E ( f ~ , ~ ) the equa t ion u 2 ~ = L n f ( u ) ~ . ~ £ ) ho lds , i . e . , the f o r m u l a JZ(~z ,~ ) is

t r u e . Cons ide r , f u r t h e r , the t r a n s f o r m a t i o n Cacti, ~ e E(..C2. ~ ) (see the p r o o f of L e m m a 2.1). Since

c,/~/~/~=/3 told ~upCc~/~f~c~,c,/~y) = ~upCc~,~)=c~, then c / ~ / ~ su~(c~,c~z~) , i.e., the formula

~[~,~) is false. By the definition of predicate P [see (4)] we conclude from what has just been stated that the relationship _ ~ ? entails the truth of the formula m(~.c~) f o r a n y _ ~ , ~ ~ i fweset ~z=~z~

and 3 ' = ~ . Conversely, let the formula ~ (~ , c~ ) be true and let ~ 2 $ , g = ~ v . As formula ~L(G$, ~ ) is true we have either $< ¢ or ~) < ~ (see Lemma 1.2). Assume that V < c~ . Then, since 2~ = sup 6~,~ ) ,

the formula /'(~),~z~ c~9 ~$. ~v ) is true and, by definition of predicate ~Z , for any we E (~:~,~) the following formula is true

f" ( w=~, u)c~, wcz~, ; c~t ,czv )

We obtah] a contradiction if we set ~ = ~2~ " Consequently, c?e ~, and, in view of the truth of formula

~C~x~,~;~#,cz~) we have -6<~'.

Second case: in the o. set (t~,~) all the exact upper bounds are trivial. Then, let ~ ~n~-(/~,~)

be m~ non t r iv ia l l ower bound. We se t ~ = % , and ~ = % . Since o~ .~]~ the f o r m u l a s ~Z C~z, ~) and 7"(K, cc) a r e t rue (the l a t t e r is t r u e by v i r tue of the condi t ions of the c a s e in ques t ion) . Now, if ~ ~ ~ then the f o r m u l a PC~.~zZ ; c ~ , ~ ) will be t rue and, consequen t ly , the f o r m u l a M ( ~ , ~ F ) will be t r u e . Con-

v e r s e l y , l e t M ( ~ . ? , ~ ) be t rue a n d l e t c~=,~ s and ~ = , ~ v . Since the f o r m u l a ,,Z(~zs,c %) is t r u e we have e i t h e r 3< ~ o r ,) -= d . A s s u m e that -) -- 5 . Then, by def in i t ion of p r e d i c a t e ./L , the f o r m u l a

(~z~, c ~ ) is t rue and, consequen t ly , the f o r n m l a 7" C~zv ,~a ) is t r ue , in con t r ad i c t i on to the f o r m u l a K ' ( ~ ,c~,c~,,~z~,~ ) being t rue and, in view of the non t r iv ia l i ty of the bound oc= Lr~j: (/3, y ) the r e l a t i o n -

s h i p s c z , ~ czo , ~ z ~ hold. Thus , d '<V , and it fol lows f r o m the t ru th of PC~z~ ~ , c ~ # , ~ ¢ ) t h a t $ , ~ [see (4)].

The l e n m m is p roven .

We now c o n s i d e r c a s e b) of T h e o r e m 2.1. Since C.Q,~) is an o. se t with non t r iv ia l exac t bounds the fol lowing a x i o m holds on the s e m i g r o u p ,~ Ct~2 _~)

(~,cy..) ~ oP) (-i 7- (o', g)). (i0)

In v iew of the e l e m e n t a r y equ iva lence of s e m i g r o u p s E(~"J,~? and E(,~73~,a), this ax iom holds on s e m i g r o u p K(ff2~ ~g). Since p r e d i c a t e ~ and, consequen t ly , a l so p r e d i c a t e F [see (6)] w e r e so def ined that the i r mean ings were one and the s a m e fo r a r b i t r a r y o. se t s , the holding of ax iom (10) on the s e m i g r o u p ~(g'2[__~) g u a r a n t e e s the p r e s e n c e of a nont r iv ia l exac t bound in the o. se t ( ~ ---~) . Consequent ly , the o. se t s C _ ~ , ~ ) and (_~ z,~) a r e on a p a r and, by v i r tue of the fo rma l na tu re of the r e l a t i onsh ip s ~-- and ~ on the r e s p e c t i v e s e m i g r o u p s _~(~ . -_~) and ~ ( K 2 , ~ ¢ ) (see L e m m a 2.3), t hese o. s e t s a r e e l e m e n t a r i l y equiva len t .

We now adduce an e xa m pl e o f two e l e m e n t a r i l y equ iva len t o. s e t s ~ , ~ ) and ( ~ ] , ~ ) e a c h of which has non t r iv i a l exac t bounds but which a re such that the s e m i g r o u p s ~ (_~, .~) and ~(~--~,~-~ ) a r e not e l e m e n t a r i l y equiva len t . This will a l so comple te the p roof of T h e o r e m 2.1 s ince the example of the f i r s t sec t ion will a l so s e r v e as m~ example i n f i r m i n g the c o n v e r s e a s s e r t i o n fo r ca se a) (see L e m m a 2.2).

174

Let (£2,,~z,), ( ~ z , ~ z ) , ( ~ , ~ f ~ be i somorph ic 1.o. se t s with card ina l i ty of the continuum with r e - • ~- ( ~ , ~ ) , spec t ive g r e a t e s t e l emen t s e~, e~, and e , . Fu r the r , le t / be an i s o m o r p h i s m of C ~ , , , ,) on

let ( £2" , ~z~'~ be a countable 1.o. set e l emen ta r i l y equivalent to ( ~ 2 , , ~ ,~ , and le t ~ ' be i ts g r e a t e s t

e lement . We a s s u m e that ~ , = ~ - - - e , e ' - - ~ ' - - = , - - z - - 6 ' , ~2, ~ £ 2 ~ {6J and _ ~ ' f 2 j - - | ~ . We set ~-J=~-2,

, ~=~2, ~rJ z Fur the r , w e s e t ~ if and only if (~,~),~z, or (._4,~)~7~ and _~'~" if

" " ~ " C 4 ; ~ 0 ~ z " forany_.4, ~ £ 2 M ~ , ~ ' ~ 2 ' . and only if (._4 , f ) ~ ~% or f , Since the l .o. se t s (g2,,~r,), (12z,~zz} , (~2/, z / ) , and (g'j',~,~') a r e pa i rwise e l emen ta r i l y equivalent , it is e a s y to build a winning s t r a t egy for p layer II in the game G,j¢~'2, ~ ) , ¢ ~ ; 4 ) ) for any rz , and this p roves that the cons t ruc ted o. se ts a re e l e - men ta r i ly equivalent [9].

We show, however , that on semigroup K(g2,~) the following ax iom is t rue :

(_.z a )(~6)(x)(HC ~)~c~d=~& (xg~ ~,H(z~)--~ ax@ ~ c~cx.m = =c )) , (ii)

but is false on semigroup S (~,~) . Indeed, if we take as ~z the transformation of _4"2 for which cz_4----/_~ when 2e ~'~2, and c~4=S-~ when M~"~; , then, obviously, ~ze ~(~ ~.) '~a-- ~z$ and, for any

or ~:~ ~2 z respectively. Thus, axiom (ii) is true on semigroup ~=(~2 ~_~),.

We now assume that this axiom is also true on semigroup E (~2/,_ --=) . Since ~x is an isotone trans- formation of (.62,~,~) ,'uld ~6' for any __~/~&'-2 / , the equation a~6,=~4, entails the equality ~z--4'

since it follows from ~z4/..~ c~6" that c~__~/~__~ ", whence ~z~z~/___~M" and, in view of the equation

~z~z2"~= ~" for any ~#-=~" we have 6~_~" . We prove analogously that from the equation ~_4,=o~8, follows the equality _~'=~/ , Thus, in axiom (II), 6----~ze, . ~Irthermore, from the fact that ~zczcz4, = ~z~,

for any _~ ~' , it readily follows that: a is a one-to-one ,napping of ~' on -4-2 ~, where =~',~'~_~'

if and only if either _~%_~/ and ~%.4~( or ~%~/ and _~/e~'2~ (otherwise, in view of the isotonicity

of ~z , it would follow from ~-~/ that ~z__~ =f, and {~--~2/, i.e., ~----~'; analogously, from ~'/--~ 2'

we obtain that .~/= ~/ which is a contradiction). Thus ~z in a one-to-one fashion maps set ~2/ with

cardinality of the continuum, onto countable set _~ . This contradiction proves the ~sity of axiom (ii)

on semigroup ~ (~,/--~) •

T h e o r e m 2.1 is comple te ly proven.

§ 3 . S e m i g r o u p s o f A l l D i r e c t e d T r a n s f o r m a t i o n s

Let C ~ ; 4 ~ be a quas io rde red se t . T r a n s f o r m a t i o n _~c of se t ~ 2 ' is cal led d i rec ted if ~ ~c~

for any ~ e ~2 r . We denote the semigroup of all d i rec ted t r a n s f o r m a t i o n s ~DC£2;,~) . We denote by

C-<-2/,,~r) the q u a s i o r d e r e d set obtained f r o m (~2,~,~) by d i scard ing all e l ements ~ for which the r e - la t ionship __44 ~ holds only when __4-=~ •

In this sect ion we shall p rove the following t h e o r e m s .

THEOREM 3 . 1 . . L e t ( ~ , - ~ ) be an o. set and ( ~ , - , ~ ) be a quas io rde red set . If the semigroups ~D(£2, ~ ) and ~ ( g ~ ; ~ ) a r e e l e m e n t a r i l y equivalent , then ( £ 2 / , ~ r ) is an o. se t e l emen ta r i l y equiv-

alent to the o. se t (=~2r,--~ r) . The conve r s e is fa l se .

T H E O R E M 3 . 2 . Let ( ~ , ~ } be an ° ' s e t with g r e a t e s t a n d l e a s t e l emen t s and let C_42/~) b e a n a r b i t r a r y quas io rde red set . If the semig roups ~D($2 _~ ) and ~ ) ( ~ 2 , ~ ) a re e l emen ta r i l y equivalent , then (A~2 z, ~ ) i s an o. se t e l emen ta r i l y equivalent to the o. se t (g2, ~ ~ The conver se is fa l se .

Before p rov ing the t h e o r e m s , we have s e v e r a l r e m a r k s to make . Fo r any quas io rde red se t ( ~ ) a n d a n y f l e . ~ I s u c h t l m t ¢ x ~ / 3 and ¢ ¢ ~ we s lml ldeno te by ~ the t r an s fo rma t ion given by the

condit ions: ~/~c¢ = /3 and cc~_4=~4, if _~ ~ x [7]. The set of all such t r an s fo rma t ions we denote by A ( $ 2 ; , ~ ) . Obviously, A ( ~ 2 , ~ ) c¢~3(_Q;,~) . We adduce he re th ree l e m m a s , p roven in [8], on the p rop -

e r t i e s of the t r a n s f o r m a t i o n s jus t in t roduced.

LEMMA 3.1. A t r a n s f o r m a t i o n c ~ e c k g ( ~ 2 ~ ) belol~gs to A(~I.~). if and only if fo rmula X ~ ) is t rue on s emig roup $D(~2~, ~ ) , where

175

and

LE_____MMA 3.__22. For any transformations cz~,~zsX~A(~2/,~ ) the equality ]3= y holds if and only

if formula Z5 C~zjs ~ , ~sj ) is true on semigroup ~9 C~ r ~ ) , where

LEMMA 3.3. For any transformations A (~2,z~&) the equality ~_-- / holds if and only if

formula K ~zp~, czs~ ) is true on semigroup , where

We denote by ¢ the equivalence re la t ion on the given q u a s i o r d e r e d se t coinciding with the i n t e r s e c - t ion of the given quas io rde r ing re la t ion and the re la t ion inve r se to i t . As is known, the or iginal quas i - o rder ing induces an o rde r ing on the co r respond ing f ac to r se t with r e s p e c t to r e l a t ion 6 . We denote this o rde r ing re la t ion by f .

We b r ing into cons idera t ion the following p red ica te

O(_,~7,y) d_/ (=g~z)( K(~,F) v K(,z,x)&'Z3(ct, y)).

Using the t r ans i t iv i ty of re la t ionship "-~ and the definit ions of p r e d i c a t e s ~s and K , we can show that p red ica te o , cons ide red on semig roup ~ 9 ( ~ 2 , ~ ) , g ives a quas io rde r ing on se t A(_z2,~). Moreove r , a lso val id is

LEMMA 3.4. Orde red set (A(_~'2,~)/~,p) is i somorph ic to the o. se t ( ~ r , ~ r ) .

P roof . It is c l e a r that fo rmulas o (x ,~ / ) and 0(5/, ~ ) a r e t rue if and only if f o rmula K (~c, 9') is t rue , because the e lements of se t A(~2,-~)/c; will be the c l a s s e s of all those t r a n s f o r m a t i o n s ~/~c, .....

c~{9,.., for which o~ . . . . . _~ = . . , . We denote the c l a s s containing ~ e ~ by ~ : ~ . We shall now

show that the one - to -one mapping / o f s e t ~2 r onsetA(-Q,~)/o,bywhich~_4=~f_ 4 for a n y ~ e ~ 2 r ,

is an i s o m o r p h i s m of (~-r,~r) and CA ~_Q,~)/c~,/,) . Obviously, the re la t ionship 6 ~ , ~ $ ) ~ p is t rue if

and only if e i the r a = $ and then ~ ~ ~$g , or if the f o rmu la S (c~$~, c~$~ ) is t rue for some ~ e _~ .

Let _4, r e £2 T . If ~ = ~ , then / _ 4 = / ~ and the re la t ionship (/:~,f~)~ p is t rue . If, however , _ ~ ' : ~ ,

then le t / ~ ~ ¢ ~ , f ? = ~ v f • S i n c e _ ~ < f ~ V , w e h a v e _4~V and, hence

(see L e m m a s 3.2 and 3.3). In o ther words , i t is t rue that ~-~c~vf , K(af$,czg4) and 2~(af_ 4 , ~ f ) ,

whence follows the t ru th of the re la t ionship (K~.~, ~ ) ~ F "

Converse ly , le t the fo rmula ( ~ f _ ~ a ~ ) e p be t rue . If ~ = ~vg" ' then 2 = _ ¢ . Let ~ 9 ~ - z ~ and

let the fo rmu la s K (~,c~ & 2, B;~c~v¢) be t rue . By L e m m a 3.3 we have ~z=~2 and f r o m L e m m a 3.2 we

obtain o~---_¢ , whence _~ . ~ (see, the definition of t r an s fo rma t ion ~z_~ ). The l e m m a is p roven .

Cons ider the axiom

( -~)(y)CX C~c) &~ X CjU) & -~dy ~ xy/~c ). (12)

LFMMA 3.5. F o r the re la t ionship ~ on set _42 z to be a n t i s y m m e t r i e it is n e c e s s a r y and sufficient that ax iom (12) be t rue on the semigroup ~D(_~2,/~,).

P roof . If r e la t ion ~.~ is a n t i s y m m e t r i c and ~c =~z~ , ~ = c ~ $ $ , a ~ a ~ , y (i.e., e i t he r ~ , o r ~ $ )

then f r o m the equat ion ~/~.ea~=c~, follow the equations ~ ¢ ~ , ~ ~ = ~ z ~ = ~ . Hence, e i ther o¢=~

176

a n d t h e n ~ 6 " - - c ? - 8 - - - a ~ . L ~ or o¢-=~ a n d t h e n a ~ 6 ~ = / s = ~ = a ~ or , f inally, oc~3)~ and then

'~'p,c 3 = ~ = ~=a2,, ~ • The f i r s t case cont radic ts the fact that cz~.. ~ a a ~ ; the second the fact that ~ is

a n t i s y m m e t r i c and also that o( ,~]s , $ ~ ~ (by the definit ion of the t r a n s f o r m a t i o n s ~r/~, ~za] ); the th i rd

case con t rad ic t s the definit ion of t r a n s f o r m a t i o n a~), . Consequently, if ~ is a n t i s y m m e t r i c the ax iom

(12) is t rue on semig roup ~(._~2,I~) •

Converse ly , if ~ .~ /~ ; / a~o~ and oc~g/~ , then t rml s fo rma t ions ~/~ and a ~ a r e different and, obvi-

ously, sa t i s fy the equat ion c,::/scz3,~=~z~/, , i .e . , ax iom (12) is fa lse on oY)('_42,t ~ ) . Thus, if ax iom (12) is

t rue on ~ ( .~2 ' ,~ ) then the re la t ion ~ is a n t i s y m m e t r i c . The l e m m a is proven.

~D ' We now turn to the proof of T h e o r e m 3.1. Let the semig roups ~ ( g 2 , ~ ) and (~,. .<) be e l e m e n - t a r i ly equivalent . By L e m m a 3.5, ax iom (12)holds on ~ ( _ ~ , ~ ) . Consequently, this ax iom holds on s e m i - groups ~ ) ( ~ 2 ; ~ ) and by L e m m a 3.5, ( ~ / ~ ) is an o .se t . F u r t h e r m o r e , s ince the par t ia l groupoids A ( ~ . ~ ) and A (A-2 '~) a r e fo rma l ly in the cor responding semlg roups (see L e m m a 3.1) they will be e l e - men ta r i ly equivalent . Hence, also e l e m e n t a r i l y equivalent will be the quas io rde red se ts (,4 692,~3, ~', ) and (A(_(2:.~),~, where % and ~z a r e the quas io rde r ings defined by p red ica te o . Then, there ex is t s a win- ning s t r e t egy for p l aye r I I in the game

for any ,z [9]. Using this s t r a t egy , and taldng L e m m a 3.4 into account, we can eas i ly cons t ruc t a winning s t r a t egy for p l aye r H in the game

. (~'2/_,~ r) a re e l emen ta r i l y equivalent . for any ,7- Consequently, the o. se ts ( ~ 2 r , ~ r) and '

In c los ing the proof of T h e o r e m 3.1 we adduce an example of two e l emen ta r i l y equivalent o. se ts

(g'2 f ~ ) m~d C_~'2 z, ~ ) such that the o. se t s ( A 2 r , ~ r) and (~2r , ~gr) a r e e l emen ta r i l y equivalent whlle the

s emig roups ~DC~2,~) aad ~DCg2,~-= ) a r e not e l emen ta r i l y equivalent .

Let A'J,', ~ , g 2 ~ be countable se t s ; l e t ~ ? be a se t with card ina l i ty of the continuum; let 6 ;&z ~. ,

and o -a be e l emen t s belonging to none of these se t s . We a s s u m e that _4~/. ~2;,_4-2f, /,~z z a r e pa i rwise

nonin te rsec t ing . We set _42~--~ff2,',Sff2 ' -4"2 z z , z

we suppose that ._4~.._4, ~ - - ~ ,~-~.. , @ ~.0", f .~ e.. , 5 -- .9 , ~'~.-~.f~, ~-.'z~ 6~, oz--~ o;, o " 4 A i ~ f " 4 e~;'.

We shal l show that the cons t ruc ted o. se t s a r e e l e m e n t a r i l y equivalent . Fo r fl~is we shall make use of the g a m m a - t h e o r e t i c a / c r i t e r ion of e l e m e n t a r y equivalence [9], cons t ruc t ing a wim~ing s t r a t egy for p l aye r II

in the game ~,, ((..42 /.~ ), (A2 z, -4 )} for any n. If on the E -th move p laye r I chose e lement ~x Y¢ e if2 ~Z • ~ .

then p l a y e r II mus t choose e lement ~c~'-~i ~ ~ 2 ~ - ~ in the following the ru les : a) if ~ c = e"~ then g

oc~-~z'=a'-~L;b) if cc.~ ' --__ @~i then oc~ -~=- /-~:E; c) if oc.<g~_¢~.~, L then a~i ~e . ' d) if

o~i = ~ e z when ¢ . - ~ then

~ i = oc, e e. C ( = ¢ . . . . . "z, ;

Consider the one - to -one {by v i r tue of d)) co r re spondence pc between the se t s [% ..... o~ i . . . . ,oct. ] and

.,oc,~] obtained after all zt moves, considering /-~ = oc~ By considering all possible 1 t case s of d ispos i t ion of the pa i r s of e l ements co i , ocj in the o. se t ( ~ ~ ~ ) one can eas i ly show, using

ru l e s a)-d) , that / is an i s o m o r p h i s m and, consequently, these ru les provide a winning s t ra tegy , i .e . , the cons t ruc ted o. se t s a re e l e m e n t a r i l y equivalent .

Fo r convenience in the ensuing d iscuss ion we introduce the following p red ica tes :

X,z c,c) d ~ C9,:z ) ( 3 ~ ) ( x c-= )&Xca)&x (8)&,SCa' ,~)&,s (.z, 8~),

Xo, c:~) ~ (.ga)Cx c~=)~ x~ Ca)& ~ (.~,,:z)),

177

Xoz (x) ~(.ea)(X(~)8,. xa~ (a)&~X~a C~=)&~(~,a)).

We use the notation ~ = | o ~} . g 2 ~ = { e ~ } . Then, the truth of formula x~.(~:) (where 6<j; ~;,]= o,¢,.z,~

on semigroup ~D(~2,£)~(g2,~)) will mean: ~---- ~z~.~ where ~ ¢ e _ ~ , / ~ e $27 (~=¢,z~ . It is obvious with this that formula x(~c) wi~ be t rue on the semigroup

f ~

if and only if one of the formulas ×g]- (~c) is t rue .

We now show that the following axiom is t rue on semigroup ~C~o, ~ )

(~a)(.,~ ¢ ~ Ca)), (13)

where

~ f £,=t"

Indeed, if we choose as 0- the t ransformat ion providing a one-to-one mapping of set J 2 f on set g ~ and

' "- " ~ , t ransformat ion leaving fixed each element X ~,_(2 A~¢ then, since ~ ~ & for any ~ . . ( 2 [ and ~ ~ ¢

belongs to semigroup ~ G ~ , ~ . Fur ther , let formula Xej(o?5~.}, where e # ¢ or ] ~ , b e t rue . Then,

~ , ~ r ¢ or / ~ ' ~ ' 2 : . If ~ ' . . (2¢ ¢ , then o.~--~ and ~z~z~=o/6~o .~ , s ince otherwise, in view of the d i rec ted- ness of ~. and o_$~, we would have ~--.j~ ,which is a contradiction. If, now, ~ e ~ f i t h e n / ~ f ' ~ , i .e. ,

= e r and ~ c~---a~ ~= e ~, but ~z~c~ ~2J , i .e. , ~_~c~ ~.' and, consequently, again a ~ ~cz . Thus, we have shown that formula ¢~ C~z) is t rue . Let h,=a~.~ , where ~ _ 4 2 , ¢, ~ e ~ 2 a ' . By definition of a

t 0 element ~=_~ belongs to _42~ . Hence the truth of formula K (aa~ ,a4 $) (see L e m m a 3.3) and, obviously,

~ a ~ = ~ i .e., formula ~z (a) is t rue . Finally, le t ~ = ,z,~ , ~=,:zl,~ , o=,:z g$ , ~ : ~ , a.r.=~. ,:z~ = ~ and

let the formulas x , z (~ )x , z ( u ) , xz. ~ (~) be t rue . Since c~ maps ~ 2 ' on g2~ in one-to-one fashion, for anY ~., gz e_4~, :~ ,~e_Q' such that ~ = Z , , ~z~z= ~'~ the equation ~ = ~ holds ff and only if the equa- tion ~'~=-~z holds. It follows f rom the equations c z ~ a, a~: ~ a and f rom the t ruth of formulas X~z,(~) ,

x,z(z.z),x~ a ( o ) that ~ = / ~ and a=3----~ because / ~ ~ (since r . -~-~z~ ~,=.zz) a n d e i t h e r A ~ p or _ ~ ~ , i .e . , e i ther 2~(z, o) or ~ ( u , o) is false (see Lemma 3.2). Thus, formula ~ (a) is also t rue and, consequently, axiom (13) is t rue on semigroup ~(_~,~-_~).

We now show that axiom (13) is false on semigroup ~)(f~,~) . Assume the converse, i .e . , that for some t ransformat ion v ~ ( ~ ~, ~ the formulas cP¢co-), cP~co_), cp~ca~ are t rue . If o_¢~ =/~ ¢oc then c z % ~ = o_ and, i nv iew of ~ Cry), the formula X z z ( ~ , ) is t rue , i .e. , ~ _ Q ~ , / s ~ . ~ z . Con-

sequently, t ransformat ion ~ leaves fixed all the e lements , with the possible exception of elements of set _42~ and, with tiffs, elements of _Q~ can, under the action of ~z , go over only into e lements of _4~ Formula % Ca) shows that for any _ ~ ~2~ the element cz~ belongs to _<~:, i .e . , ~ maps ~,-~z in ~ : . Formula % (~] a s se r t s that ~ is a one-to-one mapping of _Qz in ~ . This la t ter contradicts

the fact that ~ ) ~ a n d ~ ; have different cardinal i t ies .

Theorem 3.1 is proven.

178

We now p rove T h e o r e m 3.2. We f i r s t note that the o. se t s ( ~ f 2 [ ~ ) and ( ~ , ~ ) cons t ruc ted above have g r e a t e s t and l e a s t e l emen t s . Consequently, the example we cons ide red shows that f r o m the e l e m e n t a r y equivalence of the o. se t s (~2,~) and ('_4-2~,.4) (see the fo rmula t ion of T h e o r e m 3.2) does not follow, in genera l , the e l e m e n t a r y equivalence of s emig roups ~ ( g 2 ~ ) and ~DC~,,~4 ) . Now, le t s emig roups ~)(_c2,~)

~2 ~ and ~ ( ~ 7 g , ~ ) be e l e m e n t a r i l y equivalent . By L e m m a 3.5, ( , 4 ) i s an o. se t such that (-4~r,~r) and (~-2/, ~r ) a r e e l emen ta r i l y equivalent (see, T h e o r e m 3.1). In pa r t i cu l a r , the o. se t ( ~ 2 f , ~ ) contains a l eas t e lement . We now show that t he re is also a g r e a t e s t e l emen t in the o. set ; with this , T h e o r e m 3.2 will be comple te ly proven.

LEMMA 3.6. Let ( ~ , ~ ) be an o. se t with a l e a s t e l emen t ~ . F o r this se t to contain a g r e a t e s t e lement , i t is n e c e s s a r y and suff icient that the following ax ioms hold on semigroup ~ ( ~ , ~ ) :

C~c~ ) (x ) ( . x , ca )& x~ ca)&.. C.,~,(..v.),~ ×~,(=c).---,-..~= ~ )),

c=K...,=¢)Cx~ (&& C×(=)--.- 8(.=,~) v ~ (.,,-))),

where

(14)

(15)

x , ¢x) ~z~, cy) ( x c..=) & ( x (y) * -~ ~ (~:, 5'))),

l e t t he re be the g r e a t e s t e lement ~ . Then, for any t r an s fo rma t ion cz/~ Proof . In o. se t C-Q,~ ) ~ ( $ 2 , , ~ ) we have - 1 8 ( a ~ , a p ~ ) and ~ ( c r ~ , c ~ o ~ (see L e m m a 3.2) and, s ince ~ is a g r e a t e s t e l ement and a a l ea s t e l ement of o. se t (~'-2, ~ ) , the only t r a n s f o r m a t i o n which has this proper~y is c~# . Consequently, on ~ ( ~ 2 , -~) ax iom (14) is t rue . Fu r the r , for t r an s fo rma t ion c z ~ e i ther /~= e. and fo rmu la ×z(~z~.~) is t rue , o r / 3 - = ~ a n d t h e n fo rmula 5 ( c z ~ , ~ & p ) is t rue , i .e . , on ~b(_~2,~) ax iom (15) is t rue .

Converse ly , on semig roup ~ ( ~ , - - ~ ) l e t ax ioms (14) and (15) hold. Then, t r a n s f o r m a t i o n cc will have the f o r m ezra .where ~ is the sole max imal e l emen t of o. se t ( g 2 , ~ ) . We choose an a r b i t r a r y e l e - ment ~ ~ _42 di f ferent f r o m oc . By v i r tue of ax iom (15), for t r a n s f o r m a t i o n a j # we can find a t r a n s -

fo rmat ion ~ such that f o rm u l a X~C6) holds, and one of the fo rmulas ~(~x4a, ~), X~ C ~ ) i s t rue . Since .~=~c and ~ is the unique max ima l e l emen t in the se t ( $ 2 , . ~ ) fo rmula x~ CaSa) is fa l se . F r o m the t ru th of f o rmu la s XzC6), ~ Cc~, ~) and in view of the uniqueness of c~, follows t~e equali ty o~=cz~2 .

Whence, by the definit ion of t r a n s f o r m a t i o n ~ , we get ~ -= oc i .e . , ~c is a g r e a t e s t e lement in o. se t ('.4"2,~). Thus, the l e m m a , and with it T h e o r e m 3.2, a r e proven.

The author wishes to thank L. N. Shevrin for having posed the p r o b l e m and for help in wri t ing this

paper .

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