Galois and Post Algebras of Compositions (Superpositions)

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1 Pure ad Appled Mathematcs Joural 07; 6(): -9 do: 0.68/j.pamj IN: (Prt); IN: 6-98 (Ole) Galos ad Post Algebras of Compostos (uperpostos) Maydm Malkov Russa Research Ceter for Artfcal Itellgece, Moscow, Russa Emal address: To cte ths artcle: Maydm Malkov. Galos ad Post Algebras of Compostos (uperpostos). Pure ad Appled Mathematcs Joural. Vol. 6, No., 07, pp. -9. do: 0.68/j.pamj Receved: Jue 0, 07; Accepted: Jue, 07; Publshed: July 0, 07 Abstract: The Galos algebra ad the uversal Post algebra of compostos are costructed. The uverse of the Galos algebra cotas relatos, both dscrete ad cotuous. The foud proofs of Galos coectos are shorter ad smpler. It s oted that at-somorphsm of the two algebras of fuctos ad of relatos allows to trasfer the results of the moder algebra of fuctos to the algebra of relatos, ad vce versa, to trasfer the results of the moder algebra of relatos to the algebra of fuctos. A ew Post algebra s costructed by usg pre-teratve algebra ad by addg relatos as oe more uverse of the algebra. The uverses of relatos ad fuctos are dscrete or cotuous. It s proved that the Post algebra of relatos ad the Galos algebra are equal. Ths allows to replace the operato of cojucto by the operato of substtuto ad to exclude the operato of exst quatfer. Keywords: Fucto Algebra, Relato Algebra, Uversal Post Algebra. Itroducto The Post algebra was created by E. Post [] (9). He used ths algebra to costruct the classfcato of Boolea fuctos. The term Post algebra" appeared the frst [] (9). But the mathematcally precse defto of ths algebra was gve frst by A. Mal cev [] (976). Mal cev gave deftos of two algebras, whch he called preteratve ad teratve Post algebras. The Post algebra s the pre-teratve algebra. The teratve algebra was created by. Jabloskj ([], 958). He used t to costruct a partal classfcato of fte-valued dscrete fuctos. Therefore, he obtaed results dfferet from Post s results at costructg the classfcato of Boolea fuctos ([5], 966). The term Galos coectos" was troduced by G. Brkhoff ([6], 90). The Galos algebra of compostos was created by D. Geger [7] (968) ad by V. Bodarchuk, L. Kaluzh, V. Kotov, B. Romov [8] (969). A mathematcally precse defto of the Galos algebra was gve by. Marchekov [9] (000). The term Galos algebra of superpostos" s troduced ths artcle. Ths term s gve because ths algebra was used [6-9] to costruct the Galos coectos. Classfcato of all subalgebras of Post algebra was gve for ay k [0] (0). May statemets are gve wthout proofs or wth deas of proofs. Proofs are omtted, f they are obvous. The formalzato of the remag proofs would lead to crease the artcle volume ad to dffculty uderstadg t. ome statemets ad ts proofs are better formalzed. Notato The a symbol s used for varable values, the a character s used for a colum that cotas some varable values. The symbol s used to dex a arbtrary elemet of a set or a compoet of a sequece. The symbol s used for arty of fuctos ad relatos. The symbols f ad F are used for fuctos ad for fucto sets, the symbols r ad R are used for relatos ad for relato sets. The character s used for sorts of values of varables. Termology Predcate s a logcal formula, relato s a terpretato of a predcate by a table or matrx. Matrx les are colums of a table, matrx colums are les of a table. These matrces are used to preserve a relato by a fucto ([]): f ( a,..., a ) = a, where a s ay colum (cludg equal) of a matrx, j a j s oe of the colums of a matrx. For smplcty, the term relato table" s used stead of the term relato table terpretg a predcate". Very ofte

2 5 Maydm Malkov: Galos ad Post Algebras of Compostos (uperpostos) the terms predcate" ad relato" are detfed. The set of values of a varable s called sort. For smplcty, all varables of relatos ad fuctos have the same sort. o a set of relatos or fuctos has a sort, f these relatos or fuctos have the same sort. Relatos ad fuctos ca be ether dscrete or cotuous. They ca be tabulated. A table of a cotuous fucto s a dscrete table whch strgs are preseted by the les of odal values of varables such that the polyomal terpolato of values betwee odes s hghly accurate. Ths excludes dscotuous fuctos. However, the result obtaed for cotuous fuctos are vald for dscotuous fuctos too. The relatos ca be cotuous or partally cotuous, f the set of cotuous parts s fte or coutable.. Galos Algebra Defto. The Galos algebra R s s R s = ( R ; ζ, τ,,,&) where R s a uverse of all relatos of a sort ad ζ, τ,,, & are fudametal (or Ω ) operatos of the algebra. The sort s: N k (the frst k atural umbers), N (atural umbers), Z (tegers), Q (ratoal umbers), R (real umbers), C (complex umbers). It s geerally accepted to deote N k by k ths defto. The members of the set R are relatos ad they are deoted by r. The fudametal operatos are gve over the set R. The operatos ζ ad τ permute varables relatos. The operato reduces the umber of varables by detfyg the frst two varables. The operato s a exstece quatfer of the frst varable. The operato & s a cojucto of two relatos. A. Mal cev ([]) gave the deftos of the frst three operatos for fuctos. But they are applcable to relatos wthout chages. The operato s xr ( x,..., x ). As a result, the frst colum s removed from a table of a relato. A fucto ca be represeted by a relato ad ts frst colum ca be removed by. But the result may ot be a fucto. o the operato ca be appled oly to relatos. The cojucto of two relatos r ad r s + x + r ( x,..., x ) r ( x,..., ) where the varables x,..., x have ot equal ad varables For example, the table of real fucto x x s a table of teger dscreet fucto x x. The real fucto herts all propertes of the dscreet fucto: t belogs to maxmal cloe preservg 0 ad to mootoe cloe, the cloe of these fuctos s o-fcttous ([]) sce t has oe-membered bass { x + }, ad so o. x x +,..., x + have ot equal. But some varables from x,..., x ca be equal to some varables from x +,..., x +. I ths case, the cojucto of relatos s costructed at frst wth uequal varables, ad the the equalty of the varables s realzed. Ths cojucto s possble oly after addg the fcttous varables x +,..., x + to the relato r ad fcttous varables x,..., x to the relato r. After these addtos ad after cosequet cojucto, the relato of arty + s obtaed. The umber of les ths relato s, ad some of these les ca be equal. The frst compoets of each le r & r are oe of les of r, the followg compoets are oe of les of r. Fcttous varable s a x j for whch x x r( x,..., x, x, x,..., x ) r( x,..., x, x, x,..., x ) j j j j j+ j j j+ If = the there s a le wth ay other value x j for every le wth value x j,.e., the relato cotas all values of ts sort. A fcttous varable ca be added to a arbtrary relato by cojucto of ths relato ad the oe-ary dagoal δ (ths meas that operato of addg fcttous varable s ot prmtve,.e., the operato s ot elemetary). A dagoal of a relato r cotas oly r ( x,..., x ). The dagoal of the complete relato (the dagoal for the detcally true -ary predcate) s deoted by δ. The dagoal of the empty relato (the dagoal for the detcally 0 false predcate) s deoted by δ. The dagoal δ, ad oly t, has a fcttous varable. The other complete dagoals have the same colums. The matrx of complete dagoals cotas oly oe le, sce all les are equal. Ths le cotas all values of a sort. Cosequetly, ay fucto preserves all dagoals. Ultra-dagoal s ay dagoal δ or s cojucto of two or more complete dagoals cludg equal. Composto (superposto) s the applcato of the fudametal operatos to fuctos or relatos. The umber of all possble operatos over the set R s fte, but all of them are costructed from the fudametal operatos. Therefore, the fudametal operatos are prmtve (they caot be costructed but they ca costruct all other operatos of a algebra). It was show above that the operato of addg a fcttous varable to a relato s ot prmtve. The fudametal operatos are used to create uverses that are subsets of the uverse R. Defto. Let R be a subset of R. Uverse of R s Ultra dagoals are called dagoals [8, 9, ]. Ultra-dagoals are redudat, they are preset oly the classfcato of cocloes as a mmal cocloe.

3 Pure ad Appled Mathematcs Joural 07; 6(): -9 6 the set [ R ] cotag - members of R, - the result of applyg the operatos of permutato, detfcato ad exstece to members of [ R ], - the result of cojuctos of members of [ R ] : [ R ] = R ( g R g R ) ( g R ζg R τg R g R g R ) g, g R g & g R Ths defto s teratve. At the frst step of the terato, R cotas all members of R. At the secod step, compostos of members of R are added to R. At the ext step, the resultg set R s added by compostos of ts members. Ad so o. A uverse of fuctos s called a cloe, f t cotas the selectve (projectve) fucto (, ) = m (,..., ) = m e x x x. A cloe also cotas all selectve fuctos e x x x. A uverse of relatos s called a cocloe, f t cotas the dagoal δ. Moreover, the cocloe cotas all the dagoals of δ, sce + x x+ x x x x+ δ (,..., ) = δ (,..., ) δ (, ). A cocloe also cotas all ultra-dagoals. By the ext lemmas, a set of fuctos preservg a relato s a cloe, ad a set of relatos preserved by a fucto, s a cocloe. The proof of the lemmas s gve precse mathematcally by usg results of Mal cev ([]). Lemma. A set F = Pol( r ) s a cloe for ay relato r. e m Proof. A set Pol( r ) cotas all selectve fuctos sce em ( a,..., a ) = am, where a s ay colum from the matrx of r. Further t s prove that the set Pol( r ) s a uverse. Let f Pol( r). The f preserves a relato r : f ( a,..., a ) = aj, where a ( ) are ay colums of a matrx of r, a j s oe of the colums of ths matrx. If colums a ad a of f ( a,..., a ) are permuted the there exsts j 0 such that f ( a,..., a, a, a,..., a, a, a,..., a ) = a + + j 0 sce f preserves r. Therefore, the result of the operatos ζ ad τ over fuctos from Pol( r ) s a fucto of Pol( r ). If the frst two varables a fucto f are detfes the les wth uequal values of these varables wll be removed from the table of f. The followg les wll rema: where a s f ( a, a, a,..., a ) a wthout some les. But the fucto f cotues to preserve r after removal of some les the matrx of r. Therefore, the result of the operato s a fucto of Pol( r ). It was ecessary to prove that the result of the operato * over fuctos Pol( r ) s a fucto from Pol( r ). Let arbtrary fuctos f ad f preserve the relato r : f( a,..., a ) = a, j f ( a,..., a ) = a j, where a ad a are ay colums from the matrx of r, a ad a are oe of the colums of ths matrx. The f ( f ( a,..., a ), a,..., a ) = f ( a, a,..., a ) j j j ad there exsts j such that f( a j, a,..., a ) = a j. Lemma. A set R = Iv( f ) s a cocloe for ay fucto f. Proof. The set Iv( f ) cotas all dagoals δ ay fucto preserves ay dagoal. Further t s prove that the set Iv( f ) s a uverse., sce A fucto f preservg some relato wll preserve the relato after permutato of colums the table of ths relato,.e., after permutato of les the matrx of the relato. Therefore, the result of the operatos ζ ad τ over relatos from Iv( f ) s a relato from Iv( f ). If two varables r are detfed the les wth uequal values of the varables wll be removed from the table of r. Therefore, correspodg colums the matrx of r wll be removed. But the remaed colums have two les to be equal. The removed colums have ot these two les to be equal. o o of these colums ca be values of the fucto whe remaed colums are values of varables of the fucto. Therefore, the result of the operato over relatos from Iv( f ) s a relato from Iv( f ). The result of the operato over relatos from Iv( f ) s a relato from Iv( f ) too. Ideed, removg a colum a relato table s removg a le the relato matrx. But a fucto preservg a relato wll preserve the relato after removg oe or more les. It s eeded to prove fally that the result of the operato & over relatos from Iv( f ) s a relato from Iv( f ). + Let r = r & r ad let r ad r be relatos preserved by a fucto f : f ( a,..., a ) = aj, f ( a,..., a ) = a j, where a ad a are ay colums from matrces of r ad of r, ad where a, j a j are some colums of these matrces. A table of a relato r cotas the frst colums, each le of whch s a le from the table r. Ad the table cotas the other colums, each le of whch s a le from the table r. Hece, the matrx of the relato r cotas

4 7 Maydm Malkov: Galos ad Post Algebras of Compostos (uperpostos) colums whch frst les are les from the matrx of r ad the other les are les from the matrx of r. Hece, a fucto f preservg r ad r preserves r & r too.. Galos Coectos The set of all fuctos preservg the relato r s Pol( r ). The set of all relatos preserved by a fucto f s Iv( f ), ad Pol( R) = Pol( r), Iv( F) = Iv( f ) r R f F for the set of relatos R ad for the set of fuctos F. The Galos coecto betwee Pol( R ) ad Iv( F ) s gve by two theorems, much smpler proofs of whch are gve below. The frst Galos coecto s gve by the followg theorem. Theorem. If ad oly f a set F s a cloe the F = Pol( Iv( F )). Proof. If F = Pol( Iv( F )) the F s a cloe (lemma ). Let F be a cloe. Let R 0 be the set of all r preserved by all f F : R 0 = Iv( F ). Ad let r R0. The Pol( r ) cotas all fuctos of F ad fuctos that do ot preserve the set of the other relatos. But the the tersecto of all these fuctos s F,.e., F = Pol( Iv( F )). The secod Galos coecto s cotaed oe more theorem. Theorem. If ad oly f a R s a cocloe the R = Iv( Pol( R )). Proof. If R = Iv( Pol( R )) the R s a cocloe (lemma ). Let R be a cocloe. Let F 0 be the set of all f that preserve all r R : F 0 = Pol( R ). Ad let f F0. The Iv( f ) cotas all relatos from R ad relatos that are ot preserved by the set of other fuctos. But the the tersecto of all these relatos s R,.e., R = Iv( Pol( R )). The result of these two theorems s a very mportat Theorem. Dagrams of cloe clusos ad of cocloes clusos are at-somorphc: R R Pol( R ) Pol( R ), F F Iv( F ) Iv( F ) Proof. There s a oe-to-oe coecto betwee cloes ad cocloes. To each cloe F there correspods a cocloe R = IvF, ad to each cocloe R there correspods the same cloe F = Pol( R) = Pol( Iv( F ) (theorem ). Lkewse, to each cocloe R there correspods a cloe F = PolR, ad to each cloe F there correspods the same cocloe R = Iv( F) = Iv( Pol( R ) (theorem ). Addtoal relatos the cocloe R reduce the umber of fuctos the cloe F = Pol( R ) wth respect to the Pol( R cloe F = Pol(R ). Ideed, ) = Pol( r), ad r R addtoal r reduce F = Pol( R ) at tersecto. o R R Pol( R ) Pol( R ). Lkewse, addtoal fuctos a cloe F reduce the umber of relatos the cocloe R = Iv( F ) wth respect Iv( F to the cocloe R = Iv(F ), sce ) = Iv( f ). Ad f F F F Iv( F ) Iv( F ) The at-somorphsm of cloes ad cocloes geerates a at-somorphsm of the algebra of fuctos ad of the algebra of relatos. The ma objects of mathematcs are fuctos ad relatos that are objects of the algebra of fuctos ad the algebra of relatos. The ma problem of these algebras s a classfcato of ther objects. Ths classfcato s realzed by dagrams of clusos of cloes ad cocloes. To buld a classfcato of cloes ad the to classfy cocloes s much easer tha to do the reverse - to buld a classfcato of cocloes ad the to classfy cloes. The at-somorphsm of these algebras allows to trasfer the results of the moder algebra of fuctos to the algebra of relatos, ad to trasfer the results of the moder algebra of relatos to the algebra of fuctos.. Post Algebra The defto of the Post algebra for dscrete fuctos was gve by Mal cev ([]). Ths defto s added by cotuous fuctos ad by dscrete ad cotuous relatos. Defto. The Post Algebra P s s P s = ( P ; ζ, τ,,*) where P s a uverse of the algebra, s a sort of members of the uverse, ζ, τ, ad * are fudametal operatos of the algebra. The uverse ca cota fuctos or relatos: P = F or P = R. Frst three operatos cocde wth the frst three operatos of the Galos algebra. These operatos are applcable to fuctos wthout chages. Operato * for fuctos s substtuto f stead of the frst varable f : f * f = f x,..., x f ( x,..., x ) = = f ( f ( x,..., x ), x,..., x )) + + Operato * for relatos s substtuto r stead of the

5 Pure ad Appled Mathematcs Joural 07; 6(): -9 8 frst varable r : r where * r x,..., x + r ( r ( x,..., x ), x +,..., x + ) r ( r ( x,..., x ), x +,..., x + ) r ( x,..., x ) r ( x,..., x + ) ubsttuto of the relato r to the relato r geerates a relato of arty +. ubsttuto of a ull-ary relato to a arbtrary relato reduces arty of ths arbtrary relato by oe. A excepto s substtuto of a ull-ary relato to a ull-ary relato. The result of ths substtuto s a ull-ary relato. ubsttuto of relatos cludes substtuto of fuctos. Ideed, the substtuto operato for three-ary fuctos s f ( x, x, x) = f( f( x, x ) x ) But there s o formato about the values of the fuctos f, f ad f. Icludg ths formato, f ( x, x, x ) = x f ( f ( x, x ) = x, x ) = x s got. 0 The varable x 0 s a fcttous, sce t s ot preset the left part. The same substtuto operato for relatos s r( x, x, x, x, x ) r ( x, x, x ) r ( x, x, x ) Here fuctos f ad f are represeted by three-ary relatos r ad r. The relato r represets the fucto f after removg the fcttous varable x 0. The operato (whch adds a fcttous varable to a relato) s abset the defto of the Post algebra. Ths operato s ot prmtve, sce substtuto a two-ary relato wth two fcttous varables to ay relato adds a fcttous varable to ths relato. There are subalgebras of the Post algebra. A uverse of a subalgebra s closure of a subset of P. But fudametal operatos are equal to fudametal operatos of P. 5. Galos ad Post Algebras Are Equal The uverses of the Galos algebra ad of the Post algebra are equal for P = R. The fudametal operatos ζ, τ, are preset both algebras. The operatos &, ad the operato are dfferet. Lemma. The operatos & ad are geerated by the prmtves of the Post algebra. Proof. Frst the theorem s prove for &, ad the for. Let a cojucto be + + r ( x,..., x ) & r ( x,..., x ) It s ecessary to costruct substtuto r * r equalg the cojucto above. Let = +, let r be obtaed from r by addg a fcttous varable as the frst. Ad let =, r = r. The r ( r ( x,..., x ), x,..., x ) + + r ( x,..., x ) r ( x,..., x + + ) + + r ( x,..., x ) & r ( x,..., x ) Ideed, r has the frst varable to be fcttous before tersecto. After tersecto ths frst varable ad the last varable of r are detfed, but detfcato of ths fcttous varable ad x gves x. Ad ths detfcato decreases the umber of varables. The umber equals +. ubsttuto of ull-ary relato stead of the frst varable a arbtrary relato r mples removal of the frst varable,.e., t leads to xr ( x,..., x ). Lemma. The operato * s geerated by prmtves of the Galos algebra. Proof. If the frst varable r ad the last varable r are detfed the cojucto of r &r the r * r s got. Hece, the sets of all possble operatos both algebras are equal. Theorem. Post algebra of relatos ad Galos algebra are equal. Proof follows from lemmas ad. 6. Cocluso The Galos algebra of superpostos s costracted. The uverse of the algebra cotas both dscrete ad cotuous relatos. The fudametal operatos of ths algebra clude the cojucto ad the exstece quatfer. Much shorter ad smpler proofs of Galos coectos s foud (the proofs of Galos coectos the fudametal papers take several pages). It s oted that the at-somorphsm of clusos of cloes ad cocloes allows a laborous classfcato of relatos to replaced by a less laborous classfcato of fuctos. It s show that the ultra-dagoals are redudat ad ca be replaced by more smpler dagoals. The Post algebra has bee costructed such that the algebra has uverse cotag ether dscrete or cotuous fuctos or relatos. Almost all researchers clude the addto of fcttous varables to the Post algebra as oe more fudametal operato. It s show that ths operato s ot a prmtve. It s show that the Galos algebra ad the Post algebras of relatos are equal. But the Post algebra s smpler ad cludes the algebra of fuctos too.

6 9 Maydm Malkov: Galos ad Post Algebras of Compostos (uperpostos) Refereces [] Post E. L., Two-valued teratve systems of mathematcal logc. Prceto Uv. Press, Prceto (9). [] Rosebloom Paul C., Post algebras. I. Postulates ad geeral theory, Amer. J. Math. 6, (9). [] Mal cev A. I., Iteratve Post algebras, NGU, Novosbrsk, (Russa) (976). [] Jabloskj. V., Fuctoal costructos may-valued logcs (Russa). Tr. Mat. Ist. teklova, 5 5- (958). [5] Jabloskj,. W., Gavrlov, G. P., Kudryavcev, V. B., Boolea fucto ad Post classes (Russa), Nauka, Moscow (966). [6] Brkgoff G., Lattce theory, Amerca Math. oc., Provdece Rhode slad (90) [7] Geger D., Closed systems of fuctos ad predcates Pacfc J. Math. 7, (968). [8] Bodarchuk, V. G., Kaluzh, L. A., Kotov, V. N., Romov, B. A., Galos theory for Post algebras I II. (Russa) Kberetka,, -0 (969), 5, - 9 (969). Eglsh traslato: Cyberetcs, -5 ad 5-59 (969). [9] Marchekov C., Basc of Boolea fuctos theory (Russa), Fzmatlt, Moscow (000). [0] Malkov M. A., Classfcato of closed sets of fuctos mult-valued logc. op trasacto o appled math. : (0). [] Lau D., Fuctos algebra o fte sets, prger (006).