New problems in universal algebraic geometry illustrated by boolean equations

Transcription

1 New poblems in univesal algebaic geomety illustated by boolean equations axiv: v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic geomety and explain them by boolean equations MSC: 03G05 (boolean algebas), 03C98 (applications of model theoy). 1 Intoduction The pocess of solving equations is the cental pat of mathematics. The most geneal and impotant poblems in this aea ae the following. 1. Is a given equation consistent ove an algebaic stuctue (algeba fo shotness) A? 2. Find all solution of a given equation ove an algeba A. Thee ae many suveys and papes devoted to equations in vaious classes of algebas. Let us just mention about the suvey [1] fo goup equations. Howeve the ecent achievements of univesal algebaic geomety (see the papes[2, 3, 4] by E.Daniyaova, A.Miasnikov, V.Remeslennikov, and B.Plotkin) allow us to pose new poblems about equations (all equied definitions may be found in Section 2 of the cuent pape). 3. Systems of equations VS algebaic sets. Let Y be an algebaic set ove an algeba A. Obviously, thee exist moe than one systems of equations (systems fo shotness) with the solution set Y. Let us fix a family of systems S, and let S(Y) S beall systems withthesolution sety. Ittunsout that thenumbes S(Y) have a wide spead of values fo almost all natual S (e.g. in [5] this fact was poved fo semilattice equations). Thus, thee aises a poblem: is thee an algeba A and a natual family S such that the vaiance of the set { S(Y) } is minimal? The sense of this poblem is the following. Suppose we want to geneate andom algebaic sets ove an algeba A by a andom geneation of systems fom S. If the vaiance of the set { S(Y) } is small, the andom distibution of algebaic sets becomes close to the unifom distibution. The autho was suppoted by Russian Fund of Fundamental Reseach (poject , the esults of Sections 6,7) and Russian Science Foundation (poject , the esults of Section 4,5)

2 4. Ieducible algebaic sets. Let Y bean algebaic set ove an algeba A. Is thee an algoithm that decides whethe Y is ieducible o not? If Y is educible, can we find its ieducible components? Can you find the aveage numbe of ieducible components of all algebaic sets in A n? The impotance of this poblem is the following. Accoding to [2, 3], the stuctue of ieducible algebaic sets ove A detemines the univesal theoy of A. Moeove, if A is finite ieducible algebaic sets ove A coespond to subalgebas of A. 5. Isomophic algebaic sets. In [2] it was defined isomophisms between algebaic sets. Namely, isomophic algebaic sets have the same popeties with espect to univesal algebaic geomety. Fo any algeba A one can pose the following poblem: how many non-isomophic algebaic sets ae thee in A n? The solution of this poblem allows us to decide about the complexity of the class of all algebaic sets ove A. 6. Equationally extemal algebas. Let A n be the class of L-algebas of ode n (fo example, A n is the class of all semilattices of ode n) and S a finite set of systems. The poblem is the following: find an algeba A A n such that the numbe of consistent systems fom S is maximal (minimal) fo A. Let us efe to the papes, whee the poblems above wee solved fo some algebas. In [6] we descibe ieducible algebaic sets and compute the aveage numbe of ieducible components of algebaic sets ove linealy odeed semilattices. In [5] fo the class of semilattices of ode n it was descibed equationally extemal semilattices which have maximal (minimal) numbe of consistent equations. Above we mentioned about the pape [5], whee we conside the 3d poblem fo semilattices. The obtained esults of all papes above show that the poblems 3 6 ae nontivial even fo simple algebas. Howeve, thee exists a class of algebas, the class of boolean algebas, whee the poblems above have nice solutions. Thus, the aim of this pape is the solution of poblems 3 5 in the class of finite boolean algebas (the sixth poblem is uneasonable fo boolean algebas, since A n 1 fo any n N). So the eade may conside this pape as a vast example fo poblems above. Let us explain the plan of ou pape. In Section 2 we give basics notions of univesal algebaic geomety. Section 3 contains the ules of tansfomations of equations ove boolean algebas. Actually, any boolean system S(X) in n vaiables X can be equivalently educed to an othogonal system S (Z) in 2 n vaiables Z. Solving the 3d poblem, we pove that any algebaic set defined by a system in n vaiables is isomophic to the solution set of a unique othogonal system in 2 n vaiables. In Section 4 we descibe ieducible algebaic sets ove finite boolean algebas and decompose any algebaic set into a finite union of ieducible ones. In Section 5 we count the aveage numbe of ieducible components of algebaic sets ove finite boolean algebas. In Section 6 we give the definition of a ank of ieducibility IR(S) of a system S and count the aveage ank of ieducibility of all othogonal systems in 2 n vaiables. Thus, Sections 4 6 solve the 4th poblem fo finite boolean algebas. In Section 7 we study the 5th poblem and diectly compute the numbe of pais of isomophic algebaic sets defined by othogonal systems in 2 n vaiables.

3 2 Basic notions Let L = { (2), (2), (1),0,1} be a language of binay functional symbols, (join and meet), unay symbol (complement) and constant symbols 0, 1. Clealy, boolean algebas ae algebaic stuctues of the language L with natual intepetation of functional and constant symbols (see [7] fo moe details). Recall that fo any finite boolean algeba B thee exists a numbe 1 such that B is isomophic to the powe set algeba on elements ( B = 2 ). The numbe is called the ank of a boolean algeba B. We assume below that any boolean algeba B is nontivial, i.e. B > 1. An element a is an atom (co-atom) if {ab b B} = {0,a} (espectively, {a b b B} = {a,1}). Remak that the ank of a finite boolean algeba is equal to the numbe of atoms (co-atoms). Following [2], let us give the basic notions of algebaic geomety ove boolean algebas. Let X = {x 1,x 2,...,x n } beafiniteset of vaiables. A tem t(x) of thelanguage L is called an L-tem. The set of all L-tems in vaiables X is denoted by T L (X). A boolean equation is an atomic fomula τ(x) = σ(x) of the language L (τ,σ ae L- tems). The examples of boolean equations ae the following expessions: x i = x j, x 1 = 1, x 1 x 2 = x 3 x 4, x 1 x 2 = x 3 x 4. A system of equations (system fo shotness) is an abitay set of boolean equations. The set of all solutions (solution set) of a system S ove a boolean algeba B is denoted by V B (S). A set Y B n is algebaic ove a boolean algeba B if thee exists a system S such that Y = V B (S). A nonempty algebaic set Y is ieducible if it is not a finite pope union of othe algebaic sets. Accoding to [2], it follows that each algebaic set Y B n is decomposable into a finite union of ieducible algebaic sets Y = Y 1 Y 2... Y m (Y i Y j fo i j), (1) and the decomposition (1) is unique up to the pemutation of the sets Y i. The sets Y i in (1) ae called the ieducible components of a set Y. Let Y = V B (S) be an algebaic set ove a boolean algeba B, and S depends on vaiables X = {x 1,x 2,...,x n }. One can define an equivalence elation Y on T L (X) as follows: t(x) Y s(x) t(p) = s(p) fo each point P Y. The set of Y -equivalence classes is called the coodinate algeba of Y and denoted by Γ B (Y). By the esults of [2], it follows that Γ B (Y) is a boolean algeba and geneated by the elements x 1,x 2,...,x n. In othe wods, all coodinate algebas ae finitely geneated, and, theefoe, all coodinate algebas of algebaic sets ove boolean algebas ae finite. The following statement descibes the popeties of coodinate algebas of ieducible algebaic sets. Theoem 2.1. An algebaic set Y is ieducible ove a boolean algeba B iff Γ B (Y) is embedded into B Poof. Actually, in [2] (Theoem A) it was poved that Γ B (Y) is disciminated by B iff the algebaic set Y is ieducible. Since Γ B (Y) is finite, the discimination is equivalent to the embedding of Γ B (Y) into B.

4 Thee ae diffeent algebaic sets with isomophic coodinate algebas. Fo example, the following algebaic sets Y 1 = V B ({x 1 x 2 = x 2 }), Y 2 = V B ({x 1 x 2 = x 1 }) have isomophic coodinate algebas Γ B (Y 1 ),Γ B (Y 2 ), since the second equation above is obtained fom the fist one by the vaiable substitution (in Example 2.2 we diectly compute the coodinate algeba of the sets Y 1,Y 2 ). Following [2], an algebaic sets ae isomophic if they have isomophic coodinate algebas. Example 2.2. Let us compute the coodinate algeba of the algebaic set Y = V B (x 1 x 2 = x 2 ), whee B is an abitay nontivial boolean algeba. By the definition, Γ B (Y) is geneated by the elements x 1,x 2 (we identify hee a tem x i with its Y -equivalence class). Accoding to the axioms of boolean algebas, the equality x 1 x 2 = x 2 gives that the tem x 1 x 2 equals 0 in Γ B (Y). The diect computations give that Γ B (Y) consists of 8 elements ( Y -equivalence classes) 0,1,x 1,x 2,x 1 x 2, x 1, x 2,x 2 x 1. Theefoe, Γ B (Y) is isomophic to a boolean algeba of ank 3, and the elements x 2, x 1 x 2, x 1 (x 1, x 1 x 2, x 2 ) ae atoms (espectively, co-atoms) of Γ B (Y)). 3 Tansfomations of boolean equations Let X = {x 1,x 2,...,x n } be a finite set of vaiables. Let us define new vaiables Z = {z α α {0,1} n } indexed by all n-tuples α {0,1} n ( Z = 2 n ). Following [8], the vaiables Z ae called othogonal. By π i (α) (1 i n) we denote the pojection of a tuple α {0,1} n onto the i-th coodinate. The substitution of the vaiables X = {x 1,x 2,...,x n } is the following x i = π i (α)=1 z α. (2) Fo example, the set X = {x 1,x 2 } gives Z = {z (0,0),z (0,1),z (1,0),z (1,1) } and x 1 = z (1,0) z (1,1), x 2 = z (0,1) z (1,1). Accoding to the axioms of boolean algebas, it follows that the vaiables Z ae obtained fom X by the following ules: z α = x a 1 1 xa xa 2 n, (3) whee α = (a 1,a 2,...,a n ), a i {0,1} and x a i i = { x i if a i = 1, x i if a i = 0. (4) Foexample, thesetsx = {x 1,x 2 }, Z = {z (0,0),z (0,1),z (1,0),z (1,1) }givez (0,0) = x 1 x 2, z (0,1) = x 1 x 2, z (1,0) = x 1 x 2, z (1,1) = x 1 x 2. By (2), any system S in vaiables X = {x 1,x 2,...,x n } can be witten as S = {z α = 0 α A} α β {z α z β = 0} { α z α = 1}, (5)

5 whee A {0,1} n and α z α is the join of all vaiables z α Z (see [9] fo moe details). Moeove, in [9] it was poved that the algebaic sets V B (S ), V B (S) ae isomophic. A system of the fom (5) is called othogonal. Example 3.1. The set Y = V B (x 1 x 2 = x 2 ) (B is an abitay boolean algeba) is isomophic to the solution set of a system z (0,1) = 0, z (0,0) z (0,1) = z (0,0) z (1,0) = z (0,0) z (1,1) = z (0,1) z (1,0) = z (0,1) z (1,1) = z (1,0) z (1,1) = 0, z (0,0) z (0,1) z (1,0) z (1,1) = 1 (6) since x 1 x 2 = x 2 x 1 x 2 = 0 z (0,1) = 0. Statement 3.2. The coodinate algeba of the solution set of an othogonal system S (5) is isomophic to the boolean algeba of ank m a, whee m = Z = 2 n and a = A. Poof. Since all points P α = (p β β {0,1} n ) (α / A), { 1 if β = α, p β = 0 othewise belong to Y = V B (S), the definition of the Y -equivalence gives that the elements z α (α / A) ae nonzeo in Γ B (Y) and z α z α fo distinct α,α / A). The equations z α z β = 0 S imply that the elements z α (α / A) ae exactly the atoms of the boolean algeba Γ B (Y). Since the ank of a boolean algeba is equal to the numbe of atoms, Γ B (Y) is isomophic to the boolean algeba of ank m a. Example 3.3. Accoding to Statement 3.2, the coodinate algeba of the solution set of an othogonal system S (6) is isomophic to the boolean algeba of ank 3 (in Example 2.2 we diectly obtained the same esult). Using Theoem 2.1, we obtain that the set Y is ieducible ove any boolean algeba of ank 3. If B is the boolean algeba of ank 2 the solution set of (6) is decomposable into the union of solution sets of the following systems S 1 = S {z (0,0) = 0}, S 2 = S {z (1,0) = 0}, S 3 = S {z (1,1) = 0}. Fo the boolean algeba B of ank 2 thee ae not nonzeo elements z 1,z 2,z 3 B with z i z j = 0 (i j). Theefoe, fo any solution of S one of the following equalities holds z (0,0) = 0, z (1,0) = 0, z (1,1) = 0. Thus, V B (S) can be decomposed into a union of solution sets of S 1,S 2,S 3. Onecan pove that fo any algebaic sety B n theeexists auniqueothogonal system S in m = 2 n vaiables with the solution set isomophic to Y. Theefoe, theeaisesaone-to-onecoespondencebetweenalgebaicsetsinb n andothogonal systems in m = 2 n vaiables. It allows us below to identify the class of algebaic sets in B n and the class of all othogonal systems in m = 2 n vaiables.

6 4 Ieducible components of algebaic sets Let Y be the solution set of S (5) ove the boolean algeba B of ank. Let m = Z = 2 n, a = A. Lemma 4.1. If m a, then Y is ieducible. Poof. It diectly follows fom Statement 3.2 and Theoem 2.1. Lemma 4.2. Let m a > then Y is a union of solution sets of the following othogonal systems S B = {z α = 0 α B} α β {z α z β = 0} { α z α = 1} (7) whee B {0,1} n, B A, B = m. Moeove, the sets Y B = V B (S B ) ae ieducible components of Y. Poof. Actually, the statement of this lemma was demonstated in Example 3.3, whee the solution set of S ove the boolean algeba of ank 2 is a union of the solution sets of the systems S 1,S 2,S 3. Fo the systems S 1,S 2,S 3 the set B espectively equals {(0,0),(0,1)}, {(1,0),(0,1)}, {(1,1),(0,1)}. The poof of the lemma follows fom the statements below. 1. Let us pove Y = B Y B. Since the systems S B contain new equalities z α = 0, then obviously V B (S B ) V B (S) and B Y B Y. Let us pove the invese inclusion Y Y B. Let P = (p α α {0,1} n ) Y. Since p α p β = 0 fo all α β, then P contains at most nonzeo coodinates (and at least m zeo coodinates). Theefoe, thee exists a set B {0,1} n, B = m, B A such that p β = 0 fo all indexes β B, and theefoe P Y B. 2. Statement 3.2 implies that the coodinate algebas of algebaic sets Y B ae isomophic to B. By Theoem 2.1, all sets Y B ae ieducible. 3. Let us pove that Y B Y B fo distinct sets B,B. Let β B \B. Then the point P = (p α α {0,1} n ) with coodinates { 1 if α = β p α = 0 othewise belongs to Y B, but P / Y B. 5 Aveage numbe of ieducible components In this section we obtain a fomula fo the aveage numbe of ieducible components of algebaic sets defined by othogonal systems (5) ove the boolean algeba B of ank. Let m be the numbe of vaiables in the othogonal system S (5) and a = A. Accoding to Lemmas 4.1, 4.2, the numbe of ieducible components I(S) of the solution set of S equals { I(S) = 1 if m a ) othewise ( m a

7 The numbe of othogonal systems fo fixed m,a equals ( m a) The numbe of all othogonal systems is 2 m, theefoe the aveage numbe of ieducible components of algebaic sets defined by othogonal systems in m vaiables equals We have m 1 I = 1 2 m ( ( )( ) m m a = a ( ) m 1 m 1 m a=m m 1 ( m m a ( ) m + a m 1 ( m a )( ) ) m a. ( )( ) m m a = m a ) ( ) m 1 m ( ) m = = a ( ) m (2 m 1), and, theefoe, the aveage numbe of ieducible components is ( I = 1 m ( ) ( ) ) ( m m 2 m + (2 m 1) = 1 m ( ) ( m m a 2 m +2 m a ) ) = a=m 1 2 m Fo a fixed and m we have 1 1 ( m ) 2 m i=0 i 0 and ( ) m I 2 fo m. 6 Ranks of ieducibility a=m +1 ( 1 i=0 ( m i ) +2 m ( m ) ). Accoding to Lemmas 4.1, 4.2, the solution set of a system S (5) may be educible ove the boolean algeba of ank, but the solution set of S becomes ieducible ove the boolean algebas of highe anks. We say that a system S (5) has the ank of ieducibility IR(S) if the solution set of S is ieducible ove the boolean algeba of ank IR(S), but solution set of S is educible ove each boolean algeba of ank < IR(S) (if S is inconsistent ove any boolean algeba we put IR(S) = 0). Below we compute the aveage ank of ieducibility of othogonal systems in m vaiables. By Lemmas 4.1, 4.2, we have that IR(S) of a system S (5) equals m a, a = A. The numbe of othogonal systems in m vaiables with the ank of ieducibility m a is equal to ( m a). Theefoe, the aveage ank of ieducibility of othogonal systems in m vaiables is m ( ) ( m m ( ) m m ( ) ) m 2 m (m a) = 2 m m a = 2 m( m2 m m2 m 1) = m/2. a a a 7 Pais of isomophic algebaic sets In this section we compute the numbe of pais (Y 1,Y 2 ) such that the algebaic sets Y i ae isomophic to each othe and Y i ae defined by othogonal systems in m vaiables. Suppose algebaic sets Y i ae defined by the following othogonal systems S i = {z α = 0 α A i } α β {z α z β = 0} { α z α = 1}, (8)

8 whee A i {0,1} n The following lemma is a simple coollay of Statement 3.2. Lemma 7.1. Algebaic sets Y 1,Y 2 defined by othogonal systems S 1,S 2 (8) ae isomophic to each othe iff A 1 = A 2. Poof. In [2] (Coollay 5.7) it was poved that that Y 1,Y 2 ae isomophic iff thei coodinate algebas ae the isomophic. The application of Statement 3.2 concludes the poof. The numbe of pais (S 1,S 2 ) with A 1 = A 2 is equal to m ( )( ) ( ) m m 2m =. i i m i=0 Since thee ae exactly 2 m 2 m = 4 m pais of algebaic sets defined by othogonal systems in m vaiables, two andom algebaic sets ae isomophic with the following pobability ( 2m ) m 2 m 2 m. Applying Stiling fomula to the expession ( 2m m), we obtain that the equied pobability asymptotically equals πm 1. Refeences [1] V. A. Roman kov, Equations ove goups, Goups, Complexity, Cyptol., 4:2 (2012), [2] E. Daniyaova, A. Miasnikov, V. Remeslennikov, Unification theoems in algebaic geomety, Algeba and Discete Mathematics, 1 (2008), pp [3] E. Yu. Daniyaova, A. G. Myasnikov, V. N. Remeslennikov, Algebaic geomety ove algebaic stuctues. II. Foundations, J. Math. Sci., 185:3 (2012), [4] B. Plotkin, Seven lectues on the univesal algebaic geomety, (2002), axiv: math [math.gm]. [5] A. N. Shevlyakov, Equivalent equations in semilattices, Sib. Elekton. Mat. Izv., 13 (2016), (in Russian). [6] A. N. Shevlyakov, On ieducible algebaic sets ove linealy odeed semilattices, Goups,Complexity,Cyptology, to appea. [7] R. Bonnet, D. Monk, Handbook of Boolean Algebas, v.1-3, Elsevie (1989), 1394p. [8] S. Rudeanu, Lattice Functions and Equations, Spinge-Velag (2001),435p. [9] A. Shevlyakov, Algebaic geomety ove Boolean algebas in the language with constants, J. Math. Sci., 206:6 (2015), The infomation of the autho: Atem N. Shevlyakov Sobolev Institute of Mathematics Russia, Omsk, Pevtsova st. 13 Phone: Omsk State Technical Univesity Russia, Omsk, p. Mia, 11 a shevl@mail.u