FIRST-ORDER STRUCTURE
ETIENNE R. ALOMO TEMGOUA AND MARCEL TONGA
Received 27 September 2004 and in revised form 4 July 2005
Using-congruences and implications, Weaver (1993) introduced the concepts of pre-variety and quasipre-variety of first-order structures as generalizations of the corresponding concepts for algebras. The notion of functional completeness on algebras has been de-fined and characterized by Burris and Sankappanavar (1981), Kaarli and Pixley (2001), Pixley (1996), and Quackenbush (1981). We study the notion of functional completeness with respect to-congruences. We extend some results on functionally complete alge-bras to first-order structuresA=(A;FA;RA) and find conditions for these structures to
have a compatible Pixley function which is interpolated by term functions on suitable subsets of the base setA.
1. Introduction
Functional completeness on algebras has been studied in [2,3,4], and some results are given in [1]. Some basic notions in this field are listed in the definition below.
Definition 1.1. LetᏭ=(A;FᏭ) be an algebra and let f :A3→Abe a function.
(i) f is called amajority functionif for alla,b∈A,f(a,a,b)=f(b,a,a)=f(a,b,a)= a.
(ii) f is called aPixley functionif for alla,b∈A, f(a,b,b)= f(a,b,a)=f(b,b,a)=a. (iii) The ternary functiond:A3→Adefined byd(a,a,c)=candd(a,b,c)=aifa=b
is calledthe discriminator functiononA.
Moreover, ifᏭis a finite nontrivial algebra, thenᏭis called
(iv)primalif everyn-ary function onA,n≥1, is a term function ofᏭ; (v)quasiprimalif the discriminator function onAis a term function ofᏭ;
(vi)functionally completeif everyn-ary function onAis a polynomial function ofᏭ. Our aim here is to formulate and characterize a notion of functionally complete first-order structureA=(A;FA;RA), which takes care (in some sense) of the relations inRA.
Throughout,A=(A;FA;RA) is a nontrivial first-order structure. We denote by Con(A)
the set of congruences ofA.
Copyright©2005 Hindawi Publishing Corporation
Definition 1.2[7]. An elementθ∈Con(A) is called a-congruenceif for anym-aryrin Rand any pairsai,bi ∈θfor 1≤i≤m,a1,. . .,am ∈rAif and only ifb1,. . .,bm ∈rA.
Con(A) will denote the set of-congruences ofA; it is easy to see that Con(A)
is a sublattice of Con(A); in fact Con(A) is a complete lattice, and its largest element
denoted by 1Ais generally different fromA2= ∇A.
Let Abe the smallest congruence ofA; when A1A∇A,Ais not simple, and the
discriminator functiondonAis not a term function ofA. However,dmay be interpolated by term functions on some parts ofA.
For eachainA, the 1Aclass ofawill be denoted bya.
Definition 1.3[6]. LetAbe a structure, and let f :An→A,n≥1, be ann-ary function. (i) f is said to be 1A compatibleif 1A is a congruence of (A;f); that is, for any pairs
ai,bi ∈1Afor 1≤i≤n,f(a1,. . .,an),f(b1,. . .,bn) ∈1A.
(ii) f is said to betermal on classesif for eacha∈A, there is ann-ary termtasuch that f andtA
a coincide ona.
(iii) f is said to beterm representable on classesif there is ann-ary termtsuch that f andtAcoincide on every 1
Aclass.
(iv) LetAbe finite; thenAis said to be-primalif every 1Acompatiblen-ary function
onAis term representable on classes.
(v) LetAbe finite; thenAis said to be-quasi-primalif the discriminator function on Ais term representable on classes.
We note that a unary function f which is term representable on classes is a term func-tion.
For any elementsa,b∈A, letθ(a,b) be the principal congruence onAgenerated by a,b; ifa,b∈An, let Cong(a,b) :=
1≤i≤nθ(a(i),b(i)). For anya1,. . .,am∈
a∈Aan, letai:= a1(i),. . .,am(i)for 1≤i≤n; thenB(a1,. . .,an) := {x∈Am;x(k),x(l) ∈Cong(ak,al) for 1≤k,l≤m}is a subuniverse ofAm.
The next theorem characterizes-primality.
Theorem1.4 [6]. A finite structureAis-primal if and only if the following properties are satisfied.
(i)The only subuniverses ofA2are A,1
A, and∇A.
(ii)For any nonzero natural numbers m, n, and elementsa1,. . .,am∈
a∈Aan, ifai:= a1(i),. . .,am(i)for1≤i≤n, thenB(a1,. . .,a
n)=Sg(a1,. . .,an)as subuniverses of Am.
Condition (i) in this theorem implies thatAis minimal (i.e., it has no proper substruc-ture) and rigid (i.e., idAis the only automorphism ofA).
2.-Functionally complete structure
Given a structureA=(A;FA;RA), we will make use of the structure (of different type)
AA:=(A;FA∪ {c
We can rephraseDefinition 1.3(ii), (iii), and (iv) in terms of polynomials and obtain the following definition.
Definition 2.1. (i) f is said to bepolynomial on classesif for eacha∈A, there is ann-ary polynomialpasuch that f andpAa coincide ona.
(ii) f is said to bepolynomially representable on classesif there is ann-ary polynomial psuch that f andpAcoincide on each classa.
(iii) LetAbe finite;Ais said to be-functionally completeif anyn-ary 1Acompatible
function onAis polynomially representable on classes.
The following results give a relationship between-primality and-functional com-pleteness; the first one is a direct consequence of the definitions.
Theorem2.2. Ais-functionally complete if and only ifAAis-primal.
Theorem2.3. Ais-primal if and only if each subuniverse ofAm,m≥2, contains the set
∆A(m) := {(a,a,. . .,a) :a∈A}andAis-functionally complete.
Proof. “If ” part. Any constant function onA is a term function ofA. LetC be a sub-universe ofAm,m≥2. Then there is someu= u1,u2,. . .,u
min Am such thatu∈C. Leta∈A, the constant function with valueais representable by a termt onA. Then a,a,. . .,a = t(u1),. . .,t(um) =tA
m
(u1,. . .,um)∈C. So∆A(m)⊆C.
“only if ” part. Let f be a 1A compatiblen-ary function. Then f is representable by a polynomial p(x1,. . .,xn,ca1,. . .,cam) on classes. Let A= {b1,. . .,bk}; then ai,. . .,ai ∈ Sg(b1,. . .,bk)⊆Ak. There is a unary termt
i such thatti(bj)=ai for 1≤ j≤k. So ti representscaionA.
The termq(x1,. . .,xn)=p(x1,. . .,xn,t1(x1),. . .,tm(x1)) represents f on classes.
Now we introduce some important ideas on-functional completeness.
Lemma2.4. LetAbe a-functionally complete structure and letCbe a subuniverse of1A
which is a subdirect product ofA2such that the projectionπ
i:C→Ais not an isomorphism fori=1ori=2. Then there is somebinAsuch thaty,b ∈Cfor ally∈b.
Proof. Suppose thatπ2:C→Ais not one-to-one. Then there area1,b,a2,b ∈Csuch thata1=a2. Letnbe a natural number such that 2n≥ |b|. Let f :An→Abe a function which satisfies the following conditions.
(i) f(x1,. . .,xn)=x1ifxi,xj∈/1Afor somei < j.
(ii) f(an)⊆afor eacha∈A. (iii) f({a1,a2}n)=b.
f is 1Acompatible. Lett(x1,. . .,xn,cu1,. . .,cum) be a polynomial representing f on classes. For 1≤i≤m, letbi∈Asuch thatui,bi ∈C.
Let b:=t(b,. . .,b,b1,. . .,bm); if y∈b, then there are d1,. . .,dn∈ {a1,a2} such that y=f(d1,. . .,dn). Soy=t(d1,. . .,dn,u1,. . .,um); and
y,b=td1,. . .,dn,u1,. . .,um,tb,. . .,b,b1,. . .,bm
=tA(n+m)
d1,b,. . .,dn,b
,u1,b1,. . .,um,bm
∈C. (2.1)
Lemma 2.5. If Ais -functionally complete and minimal, then every subuniverse of 1A
either is the graph of an automorphism ofAor contains Aproperly.
Proof. LetCbe a subuniverse of 1A. SinceAis minimal,Cis a subdirect product ofA2.
Suppose thatCis the graph of a permutationαofA.
Let f be an n-ary operation of A; then a1,α(a1),. . .,an,α(an) ∈C implies that f(a1,. . .,an),f(α(a1),. . .,α(an)) ∈C, thus f(α(a1),. . .,α(an))=α(f(a1,. . .,an)). Also, let r be anm-ary relation of A; sinceC is a subuniverse of 1A, for any elements a1,
α(a1),. . .,am,α(am)∈C, (a1,. . .,am)∈rif and only if (α(a1),. . .,α(am))∈r. Therefore αis an automorphism ofA.
IfCis not the graph of a permutation ofA, it contains two elementsa1,banda2,b witha1=a2, or two elementsa,b1 anda,b2 withb1=b2. Then fromLemma 2.4, there is somebsuch thatb× {b} ⊆C; sob,b ∈C. SinceAhas no proper subuniverse,
Ais contained inC.
Theorem2.6. IfAis-functionally complete and minimal, then there is a unary termt such that for eacha,b ∈t(1A), there is an automorphismσofAsuch thatσ(a)=b.
Proof. LetB= {|t(A)|; tan unary term}; thenB⊆N∗. Letn0=min(B) and lett0 be a unary term with|t0(A)| =n0. Leta,b ∈t0(1A); thenC=Sg(a,b) is a subuniverse of
1A. FromLemma 2.5,Cis the graph of an automorphism ofAor Ais a proper subset
ofC.
If AC, then there is a unary termt1such thatt1(a)=t1(b); thus|t1t0(A)|< n0, a contradiction. ThereforeCis the graph of an automorphismσofA, andσ(a)=b.
Corollary2.7. IfAis-functionally complete and minimal, and there is a unary term tanda∈Asuch thatt(A)⊆a; then there is a unary termt0such that for allb,c∈t0(A), there is an automorphismσofAsuch thatσ(b)=c.
Proof. LetB:= {|t(A)|;ta unary term andt(A)⊆a}; then the result follows by using the
same argument as in the proof ofTheorem 2.6.
3. Interpolation of Pixley functions
In this section, we examine some links between-functional completeness and term interpolation of 1Acompatible Pixley functions.
Theorem3.1. IfAis-functionally complete and minimal, then for eacha∈A, there is a 1Acompatible Pixley function which is representable by a term ona.
Proof. Letd∈Abe fixed, and letpbe the ternary function defined by
p(a,b,c)=
a ifa=b=candc∈ {a,b}or (a=b=c), c if (a=b=canda=bora∈ {b,c}andb=c, b elsewhere.
(3.1)
pis a 1Acompatible Pixley function. Sopis representable by a polynomialp0(x,y,z) :=
of A generated by e=t0(d) is A. There are m unary terms t1,. . .,tm such that a1= t1(e),. . .,am=tm(e). Sop0(x,y,z)=t(x,y,z,t1(e),. . .,tm(e)).
The term s(x,y,z,w) :=t(x,y,z,t1(w),. . .,tm(w)) is a 4-ary term and p(x,y,z)= s(x,y,z,e) on classes. Consider the termm(x,y,z) :=s(x,y,z,t0(x)) and b,c∈d; then t0(b),e ∈t0(1A); fromTheorem 2.6there is an automorphismσ ofAsuch thatt0(b)= σ(e). Then|σ(b)| = |σ(b)|; so,
m(b,b,c)=sb,b,c,t0(b)=sb,b,c,σ(e)=σsσ−1b,σ−1b,σ−1c,e; (3.2)
that is
m(b,b,c)=σ pσ−1b,σ−1b,σ−1c=σσ−1(c)=c. (3.3)
Similarly,m(b,c,b)=bandm(b,c,c)=b. Somis a ternary term which is Pixley’s on
d.
Corollary3.2. IfAis-functionally complete and minimal, and there is a unary term t(x)anda∈Asuch thatt(A)⊆a, thenAhas a1Acompatible Pixley function which is term
representable on classes.
The proof is similar to the proof ofTheorem 3.1, using the unary termt0ofCorollary 2.7.
A first-order structureAis-arithmeticalif Con(A) is arithmetical.
LetKbe a class of first-order structures, and consider the following classes. (i)H∗(K) is the class of-quotients of structures inK.
(ii)S(K) is the class of substructures of structures inK. (iii)P(K) is the class of products of structures inK.
(iv) A-variety is a class of structures preserved byH∗,S, andP. SoH∗SP(K) is the -variety generated byK.
It is proved in [7] that a-varietyᐂis-arithmetical if and only if there is a ternary termq(x,y,z) such that for anyAinᐂanda,b∈A,a,b ∈1Aimplies thatqA(a,b,a)=
qA(b,b,a)=qA(a,b,b)=a.
This result says that there is a term which is Pixley’s on 1Aclasses for eachAinᐂ. The
remark below gives some criteria for a-variety generated by a-functionally complete structure to be-arithmetical.
Remark 3.3. Suppose thatAis-functionally complete, minimal, and there is a unary termtanda∈Asuch thatt(A)⊆a.
If for eachB∈H∗SP(A), the ternary termm(x,y,z) ofCorollary 3.2is a Pixley func-tion on 1Bclasses, thenH∗SP(A) is-arithmetical; in particular, if 1B⊆(1A)Ifor each
non empty setIand eachB⊆AI.
Proof. Letq:A3→Abe the function defined by
q(a,b,c)=
a ifa=b=c,a=b,
c if not. (3.4)
qis 1A compatible. Soqis representable on classes by a polynomialt(x,y,z,cb1,. . .,cbk). Lett0be the unary term ofCorollary 2.7ande∈t0(A); then Sg(e)=A. There arekunary termst1,. . .,tk such thatb1=t1(e),. . .,bk=tk(e). So q(x,y,z)=t(x,y,z,t1(e),. . .,tk(e)) on classes. Consider the termm(x,y,z) :=t(x,y,z,t1(t0(x)),. . .,tk(t0(x))) anda,b,c∈A such thata=b=c. Then there is an automorphismσofAsuch thatt0(a)=σ(e).
If a=b, then m(a,b,c) =t(a,b,c,t1(t0(a)),. . .,tk(t0(a))), that is, m(a,b,c)= σq(σ−1a,σ−1b,σ−1c)=σσ−1(a)=a.
Similarly, ifa=b, thenm(a,b,c)=σq(σ−1a,σ−1b,σ−1c)=σσ−1(c)=c.
Som(x,y,z) represents the discriminator function on classes; andAis-quasiprimal. Theorem3.5. IfAis-functionally complete, minimal, and there is a unary termtsuch that|t(A)| =1, thenAis-primal.
Proof. UsingTheorem 2.3, we will show that any subuniverse ofAm contains A(m). Lett(A)= {a}; we have Sg(a)=A. LetCbe a subuniverse ofAmandu∈C;a,. . .,a = tAm
(u)∈C. So A(m)=Sg(a,. . .,a)⊆CbecauseAis minimal.
4. Some examples of-functionally complete structure
We begin with a version of a Baker-Pixley lemma (see [1, Section IV-10]) suitable for our purpose.
Lemma4.1. LetAbe a finite first-order structure such that there is a majority function which is term representable on classes and let f :An→Abe ann-ary function,n≥1.
If for each nonzero natural number m≤ |A/1A|+ 1, and any elements a1,. . .,am in
a∈Aan,B=Sg(a1(1),. . .,am(1),. . .,a1(n),. . .,am(n))is preserved by f, then there is a termp(x1,. . .,xn)representing f on classes.
The proof is similar to the proof of the original lemma in [1].
Example 4.2. LetZ=(Z; +,−, 0;≤) be the ordered group of integers. Letθbe an equiva-lence relation onZ;θis a congruence if and only if there is a natural numbernsuch that a,b ∈θif and only ifa−b∈nZ.
Claim (i). We have 1Z= Z.
Proof. Letθbe an equivalence relation onZsuch thatθ= Z. Then there is somea,b ∈
θsuch thata=b. Thusa < borb < a. By symmetry, we can consider only the first case; sincea,b,b,a ∈θ,a≤bandba,θis not a-congruence. So Z is the only
-congruence onZ.
Claim (ii). Ais-functionally complete. Proof. Lethbe ann-ary function on{0, 1},n≥1.
Ifh(0,. . ., 0)=0 andh(1,. . ., 1)=0, thenhis representable on classes by the polyno-mialc0.
Ifh(0,. . ., 0)=0 andh(1,. . ., 1)=1, thenhis representable on classes by the first pro-jection.
Ifh(0,. . ., 0)=1 andh(1,. . ., 1)=0, thenhis representable on classes by the polyno-mialt(x1,. . .,xn) :=x1+c1.
Ifh(0,. . ., 0)=1 andh(1,. . ., 1)=1, thenhis representable on classes by the polyno-mialc1.
Thereforehis polynomially representable on classes.
Since{0}is a subuniverse theA,Ais not-primal. The functionmdefined onAby
m(a,b,c)=
a ifa=bora=c, b ifb=c,
c elsewhere
(4.1)
is not a polynomial function ofᏭ=(A; +,−, 0). SoᏭis not functionally complete.
Example 4.3. Consider the setA= {a,b,c}and the operations f,gonAdefined as fol-lows:
f(x,y,z,u)=
d(x,y,z) ifu=aand
(x,y,z∈ {b,c}) or (x=a=y) or (y=a=x),
a if not.
(4.2)
g(a)=a,g(b)=c, andg(c)=b.
Consider the structure A=(A;f,g,a;r), wherer= {a,b,a,c,b,a,c,a}. It is easy to see that Con(A)={ A,θ,∇A}, whereθ= A∪ {b,c,c,b}. Sincea,a,b,a ∈ ∇A,a,b ∈randa,a∈/ r,∇Ais not a-congruence. We can easily verify thatθ is a -congruence; thus 1A=θ, and|A/θ| =2.
We prove thatAAis-primal.
We have Con(AA)=Con(A); so 1AA=θ. The term f(x,y,z,ca) represents the discrim-inator function on classes. Lethbe ann-ary function onA,n≥1, which preservesθ; using
Lemma 4.1, we will prove that for each nonzero natural numberm≤3, for all elements a1,. . .,amina∈Aan,hpreserves the subuniverseB=Sg(a1(1),. . .,am(1),. . .,a1(n),. . ., am(n)) of (AA)m.
IfBis not a subset ofθ, then there isx,y ∈Bsuch thatx,y∈/ θ; sinceg preserves B, we have{a,b,a,c} ⊆Bor{b,a,c,a} ⊆B. By symmetry, we can consider only the first case.
Since b,c = f(b,b,a,b,a,c,a,a)∈B, c,b =g(b,c)∈B, and b,a = f(b,b,a,b,a,a,a,a)∈B, soc,a =g(b,a)∈B; thereforeB=A2.
If AB⊆θ, then there isx,y ∈Bsuch thatx=y; sox,y = b,corx,y = c,b. Sinceb,c ∈Bif and only ifc,b ∈B, we haveB=θ. Therefore,Bis one of the subuniverses A,θ, and∇A. ThushpreservesB.
(iii) Ifm=3, for 1≤i≤3, we denote byaithe set{ai(1),. . .,ai(n)}. Ifa1 = {a},a2 = {a}, anda3 = {a},B= A(3) andhpreservesB.
Ifa1={a},a2={a}, anda3 = {b}, thena,a,b ∈B; soa,a,c =g(a,a,b)∈ B, and b,b,a = f(b,b,b,a,a,b,a,a,a,a,a,a) ∈ B; moreover, c,c,a = g(b,b,a), b,b,c = f(b,b,b,a,a,b,a,a,c,a,a,a), andc,c,b =g(b,b,c) are inB; thusB= A×AandhpreservesB.
Ifa1 = {a},a2 = {a}, anda3 = {c}, thenB= A×A. Ifa1 = {a},a2 = {a}, anda3 = {b,c}, thenB= A×A.
Ifa1={a},a2={b}, anda3={c}, thena,b,c ∈B, anda,c,b =g(a,b,c)∈ B, b,a,a = f(b,b,b,a,b,c,a,a,a,a,a,a)∈B, and c,a,a =g(b,a,a)∈B; a,b,b = f(b,b,b,b,a,a,a,a,a,a,a,a)∈B, anda,c,c ∈B;b,c,c=f(b,b,b, a,b,b,a,c,c,a,a,a)∈B, and c,b,b ∈B; b,c,b = f(b,b,b,a,b,c,b,c,c, a,a,a)∈B, and c,b,c ∈B;b,b,c = f(b,b,b,a,b,b,c,b,c,a,a,a)∈B, and c,c,b ∈B.
ThusB= {(x,y,z)∈A3;y,z ∈θ}. SohpreservesB.
Letεbe a permutation of{1, 2, 3}; the functionαε:A3→A3defined byαε(x1,x2,x3)= (xε(1),xε(2),xε(3)) is an automorphism ofA3.
Therefore the subuniverseBis one of the elements of the setE, whereE={αε({ A(3), A×A, A×θ}); εa permutation of{1, 2, 3}}. ThushpreservesB. Sohis term repre-sentable on classes andAAis-primal.
The setB:= {a,a,a,b,a,c}is a subuniverse ofA2; thusAis not-primal. We prove thatD:= {a,a,b,b,c,a,a,a,c,c,b,a,a,a,a}is a subuniverse ofA3. Ifx∈ {a,a,b,a,a,c,a,a,a}and y,u,vare in D, then f(x,y,u,v)∈ {a,a,b, a,a,c,a,a,a} ⊆D.
Ifx,vare in{b,c,a,c,b,a}andy,uare inD, then f(x,y,u,v)∈ {a,a,b,a,a,c, a,a,a} ⊆D.
Ifx= b,c,a,y∈ {a,a,b,a,a,c},u∈D, andvis an element of{a,a,b,a,a,c, a,a,a}, then f(x,y,u,v)= b,c,a ∈D.
If x = b,c,a, y= b,c,a, u ∈ {b,c,a,c,b,a}, and v ∈ {a,a,b,a,a,c, a,a,a}, then f(x,y,u,v)∈ {b,c,a,c,b,a} ⊆D.
Ifx= b,c,a,y= c,b,a, andu,v∈ {a,a,b,a,a,c,a,a,at}, then f(x,y,u,v)∈ {a,a,b,a,a,c,a,a,a} ⊆D.
Ifx= b,c,a,y= c,b,a,u∈ {b,c,a,c,b,a}, andv∈ {a,a,b,a,a,c,a,a,a}, then f(x,y,u,v)= b,c,a ∈D.
The functionh:A2→Adefined by
h(x,y)=
y ifx=y,
b if (x=aandy=b) or (x=candy=a), c elsewhere
(4.3)
isθcompatible. The subuniverses ofA2are A,C= {b,c,a,a,c,b},θ,{a} ×A,A× {a}andA2; sohpreserves the subuniverses ofA2. Sinceb,c,c =h(a,a,b,b,c,a)∈/ D, using [1, Lemma IV-10.4], we see that there is no majority term forA.
The algebraA=(A;f,g,a) is not primal since{a}is a subuniverse ofA. Con(A)= { A,θ,∇A}, thusAis not quasiprimal and is not functionally complete.
Acknowledgment
We would like to thank the referees for pointing out several mistakes in the first version of the work.
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Etienne R. Alomo Temgoua: Department of Mathematics, ´Ecole Normale Sup´erieure, University of Yaound´e-1, P.O. Box 47, Yaound´e, Cameroon
E-mail address:[email protected]
Marcel Tonga: Department of Mathematics, Faculty of Science, University of Yaound´e-1, P.O. Box 812, Yaound´e, Cameroon
