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Graphs of finite algebras, edges, and connectivity 6 Andrei A.Bulatov 1 0 2 n Abstract a J We refineandadvancethe studyofthe localstructureofidempotentfi- 9 nitealgebrasstartedin[A.Bulatov,TheGraphofaRelationalStructureand Constraint Satisfaction Problems, LICS, 2004]. We introduce a graph-like ] O structureonanarbitraryfiniteidempotentalgebraomittingtype1. Weshow that this graph is connected, its edges can be classified into 3 types corre- L spondingtothelocalbehavior(semilattice,majority,oraffine)ofcertainterm . s operations, and that the structure of the algebra can be ‘improved’without c introducingtype1bychoosinganappropriatereductoftheoriginalalgebra. [ Thenwerefinethisstructuredemonstratingthattheedgesofthegraphofan 1 algebracanbemade‘thin’,thatis,therearetermoperationsthatbehavevery v similartosemilattice,majority,oraffineoperationson2-elementsubsetsof 3 0 thealgebra. Finally, we provecertainconnectivitypropertiesof therefined 4 structures. 7 This research is motivated by the study of the Constraint Satisfaction 0 Problem,althoughtheproblemitselfdoesnotreallyshowupinthispaper. . 1 0 6 1 Introduction 1 : v The study of the Constraint Satisfaction Problem (CSP) and especially the Di- i X chotomy Conjecture triggered a wave of research in universal algebra, as it turns r outthatthealgebraicapproachtotheCSPdevelopedin[15,20]isthemostprolific a one inthis area. Thesedevelopments have led toanumber ofstrong results about the CSP, see, e.g., [1, 4, 5, 8, 10, 12, 14, 19]. However, successful application of the algebraic approach also requires new results about the structure of finite algebras. Two ways to describe this structure have been proposed. One is based on absorptionproperties[2,3]andhaslednotonlytonewresultsontheCSP,butalso tosignificantdevelopments inuniversalalgebraitself. In this paper we refine and advance the alternative approach originally devel- oped in [7, 11, 16], which is based on the local structure of finite algebras. This approach identifiessubalgebras orfactors ofanalgebra having ‘good’termopera- tions,thatis,operationsofoneofthethreetypes: semilattice,majority,oraffine. It 1 thenexploresthegraphorhypergraphformedbysuchsubalgebras, andexploitsits connectivityproperties. Inanutshell,thismethodstemsfromtheearlystudyofthe CSPoversocalledconservativealgebras[10],andhasledtoamuchsimplerproof ofthedichotomyconjectureforconservativealgebras[13]andtoacharacterization of CSPssolvable by consistency algorithms [9]. In spite of these applications the original methods suffers from a number of drawbacks that make its use difficult. Inthepresentpaperwerefinemanyoftheconstructions andfixthedeficienciesof theoriginal method. Asin[7,16]anedgeisapairofelementsa,bsuchthatthere is afactor algebra of the subalgebra generated bya,b that has an operation which issemilattice,majority,oraffineontheblockscontaininga,b;thisoperationdeter- minesthetypeofedgeab. Inthispaperweallowedgestohavemorethanonetype ifthereareseveralfactorswitnessingdifferenttypes. Themaindifferencefromthe previous results is the introduction of oriented thin majority and afiine edges. An edge ab is said to thin if there is a term operation that is semilattice on {a,b}, or thereisatermoperationthatsatisfiestheidentitiesofamajorityoraffineterm(say, in variables x,y) on {a,b}, but only when x = a and y = b. Oriented thin edges allow us to prove a stronger version of the connectivity of the graph related to an algbera. Thisupdated approach makes itpossible togiveamuchsimpler proof of theresultof[9](seealso[4]),however,thisisasubject ofsubsequent papers. 2 Preliminaries Interminology andnotation wefollowthestandard textsonuniversal algebra[17, 22]. We also assume familiarity with the basics of the tame congruence theory [18]. Allalgebras inthis paper are assumed tobefinite, idempotent, and omitting type1. Algebras willbe denoted byA,B, etc. Thesubalgebra of an algebra A generated by a set B ⊆ A is denoted Sg (B), or if A is clear from the context simply A by Sg(B). The set of term operations of algebra A is denoted by Term(A). Sub- algebras ofdirect products areoften considered as relations. Anelement (atuple) of A ×···×A isdenoted inboldface, say, a, and itsith component isreferred 1 n to as a[i], that is, a = (a[1],...,a[n]). The set {1,...,n} will be denoted by [n]. For I ⊆ [n], say, I = {i ,...,i }, i < ··· < i , by pr a we denote the 1 k 1 k I k-tuple (a[i ],...,a[i ]), and forR ⊆ A ×···×A by pr R wedenote the set 1 k 1 n I {pr a | a ∈ R}. If I = {i} or I = {i,j} we write pr ,pr rather than pr . I i ij I The tuple pr a and relation pr R are called the projections of a and R on I. A I I subalgebra (a relation) R of A × ··· × A is said to be a subdirect product of 1 n A ,...,A if pr R = A for every i ∈ [n]. Fora congruence α of A and a ∈ A, 1 n i i by aα wedenote the α-block containing a, and byA/ the factor algebra modulo α 2 H thick edge thin edge classes of θ ab a b <a,b> Figure1: Edgesandthickedges α. ForB ⊆A2,thecongruencegeneratedbyB willbedenotedbyCgA(B)orjust Cg(B). By0 ,1 we denote the least (i.e. the equality relation), and the greatest A A (i.e.thetotalrelation)congruenceofA,respectively. Again,weoftensimplifythis notation to0,1. 3 Graph: Thick edges 3.1 The three types ofedges Let A be an algebra with universe A. We introduce graph G(A) as follows. The vertex set is the set A. A pair ab of vertices is a edge if and only if there exists a congruenceθofSg(a,b)andatermoperationofAsuchthateitherf/ isanaffine θ operation on Sg(a,b)/ , or f/ is a semilattice operation on {aθ,bθ}, or f/ is a θ θ θ majorityoperation on{aθ,bθ}. Ifthereexistsacongruence andatermoperation ofAsuchthatf/ isasemi- θ latticeoperationon{aθ,bθ}thenabissaidtohavethesemilatticetype. Anedgeab isofmajority typeifthereareacongruence θ andf ∈ Term(A)(atermoperation of A, respectively) such that f/ is a majority operation on {aθ,bθ}. Finally, ab θ hasthe affine typeifthere areacongruence θ andf ∈ Term(A)(aterm operation ofA,respectively) such thatf/ isanaffineoperation onha,bi/ . Inallcases we θ θ saythatcongruence θ witnessesthetypeofedgeab. Note that, for every edge ab of G(A), there is the associated pair aθ,bθ from the factor structure. We willneed both of these types of pairs and willsometimes callaθ,bθ athickedge(seeFig.1). Thesmallestcongruence certifying thetypeof anedgeabwillbedenotedbyθ . ab Note also that a pair ab may have more than one type witnessed by different congruences θ. Sometimes weneed astricter version oftype. A pair abis strictly semilattice ifitissemilattice; abissaidtobestrictly majority, ifitismajority but not semilattice. Finally, pair ab is said to be strictly affine if it is affine, but not semilattice ormajority. 3 3.2 General connectivity Theorem1 If an idempotent algebra A omits type 1, then G(B) is connected for everysubalgebra ofA. Let A = (A;F) be an idempotent algebra. Recall that a tolerance of A is a binary reflexive and symmetric relation compatible with A. Thetransitive closure ofatolerance isacongruence ofA. Inparticular, ifAissimplethen thetransitive closureofeveryitstolerancedifferentfromtheequalityrelationisthetotalrelation. If a tolerance satisfies this condition then we say that it is connected. Let τ be a tolerance. A set B ⊆ Amaximal with respect of inclusion and such that B2 ⊆ τ issaidtobeaclassofτ. Wewillneedthefollowingsimpleobservation. Lemma2 Everyclassofatoleranceofanidempotentalgebra isasubalgebra. Let G = (V,E) be a hypergraph. A path in G is sequence H ,...,H of 1 k ∅ hyperedges such that H ∩H 6= , for 1 ≤ i < k. The hypergraph G is said i i+1 tobeconnected if,foranya,b ∈ V,thereisapathH ,...,H suchthata ∈ H , 1 k 1 b ∈H . k Clearly, the universe of an algebra A along with the family of all its proper subalgebras forms a hypergraph denoted by H(A). Lemma 2 implies that, for a simple idempotent algebra A, the hypergraph H(A) is connected unless A is tolerance free. Inthelattercaseitcanbedisconnected. IfαisacongruenceofafinitealgebraAandRisacompatiblebinaryrelation, thentheα-closureofRisdefinedtobeα◦R◦α. Arelationequaltoitsα-closure is said to be α-closed. If (α,β) is a prime quotient of A, then the basic tolerance for (α,β) (see [18], Chapter 5) is the α-closure of the relation α∪ {N2 | N is an (α,β)-trace} if typ(α,β) ∈ {2,3}, and it is the α-closure of the compatible S relationgeneratedbyα∪ {N2 |N isan(α,β)-trace}iftyp(α,β) ∈ {4,5}. The basictolerance isthesmallestα-closedtolerance τ ofAsuchthatα 6= τ ⊆ β. S Let (α,β) is a prime quotient of A. An (α,β)-quasi-order is a compatible reflexiveandtransitiverelationRsuchthatR∩R−1 =α,andthetransitiveclosure of R ∪ R−1 is β. The quotient (α,β) is said to be orderable if there exists an (α,β)-quasi-order. By Theorem 5.26 of [18], (α,β) is orderable if and only if typ(α,β) ∈{4,5}. Recall that an element a of an algebra A is said to be absorbing if whenever t(x,y ,...,y )isan(n+1)-arytermoperationofAsuchthattdependsonxand 1 n (b ,...,b ) ∈ An, then t(a,b ,...,b ) = a. A congruence θ of A2 is said to be 1 n 1 n skew if it is the kernel of no projection mapping of A2 onto its factors. If A is a simple idempotent algebra, then the result of [21] states that one of the following holds: (a)Aistermequivalenttoamodule;(b)Ahasanabsorbing element;or(c) A2 hasnoskewcongruence. 4 Wealsoneedthefollowingeasyobservation. Lemma3 Let R be an n-ary compatible relation on A such that, for any i ∈ {1,...,n}, pr R = A. Then, for any i ∈ [n], the relation tol = {(a,b) | i i therearea ,...,a ,a ,...,a ∈Asuchthat(a ,...,a ,a,a ,...,a ), 1 i−1 i+1 n 1 i−1 i+1 n (a ,...,a ,b,a ,...,a )∈ R}isatoleranceofA. 1 i−1 i+1 n Toleranceoftheformtol willbecalledlinktolerance, orithlinktolerance i Proposition 4 LetAbeasimpleidempotent algebra. (1)Iftyp(A)∈ {4,5}thenH(A)isconnected. (2)Iftyp(A)= 3andAhasapropertolerance, thenH(A)isconnected. (3)Iftyp(A)= 2thenAistermequivalent toamodule. (4) If A = Sg(a,b), typ(A) = 3 and A is tolerance free, then either a,b are connected inH(A),orthereisabinary termoperation f oraternary termoperation g such that f is a semilattice operation on {a,b}, or g is a majority operation on {a,b}. Proof: (1) By Theorem 5.26 of [18], there exists (0,1)-quasi-order ≤ on A, which is, clearly, just a compatible partial order. Let a ≤ b ∈ A be such that a ≤ c ≤ b implies c = a or c = b. We claim that {a,b} is a subalgebra of A. Indeed, for any term operation f(x ,...,x ) of A and any a ,...,a ∈ {a,b}, 1 n 1 n we have a = f(a,...,a) ≤ f(a ,...,a ) ≤ f(b,...,b) = b. Finally, it follows 1 n fromLemma5.24(3)andTheorem5.26(2)that≤isconnected. (2)FollowsstraightforwardlyfromLemma2andthefactthatthetransitiveclosure ofanypropertolerance ofAisthetotalrelation. (3)Followsfromtheresultsof[21]. (4)Weconsidertwocases. CASE 1. Thereisnoautomorphism ϕofAsuchthatϕ(a) = bandϕ(b) = a. Consider the relation R generated by (a,b),(b,a). Bythe assumption made, R is not the graph of a bijective mapping. By Lemma 3, tol ,tol are tolerances of A 1 2 different from the equality relation. Thus, they are the total relation. Therefore, there is c ∈ A such that (a,c),(b,c) ∈ R. If both Sg(a,c),Sg(b,c) are proper subalgebras ofA,thena,bareconnected inH(A). Otherwise, let,say, Sg(a,c) = A. Since (b,a),(b,c) ∈ R and A is idempotent, (b,d) ∈ R for any d ∈ A. In particular, (b,b) ∈ R. Thismeansthatthereisabinary termoperation f suchthat f(a,b) = f(b,a) = b,asrequired. 5 CASE 2. Thereisanautomorphism ϕofAsuchthatϕ(a) = bandϕ(b) = a. Consider the ternary relation R generated by(a,a,b),(a,b,a),(b,a,a). Asinthe previous case, if we show that (a,a,a) ∈ R, then the result follows. Let also S = {(c,ϕ(c)) | c ∈ A}denotes the graph of an automorphism ϕ with ϕ(a) = b andϕ(b) = a. CLAIM 1. pr R = A×A. 1,2 LetQ = pr RandQ′ = {(c,ϕ(d)) |(c,d) ∈ Q}. SinceQ′(x,y) = ∃z(Q(x,z)∧ 1,2 S(z,y)),thisrelationiscompatible. Clearly,Q = A×AifandonlyifQ′ = A×A. Notice that (a,a),(b,b),(a,b) ∈ Q′. Since typ(A) = 3 and A is tolerance free, every pair c,d ∈ A is a trace. Therefore, there is a polynomial operation g(x) with g(a) = c,g(b) = dand, hence, there is a term operation f(x,y,z) such that f(a,b,x) = g(x). Forthisoperation wehave a b a c f , , = ∈ Q′. a b b d (cid:18)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19) Next we show that tol cannot be the equality relation. Suppose for contra- 3 diction that it is. Then the relation θ = {((c ,d ),(c ,d )) | there is e ∈ A 1 1 2 2 such that (c ,d ,e),(c ,d ,e) ∈ R} is a congruence of Q = A2. It cannot be a 1 1 2 2 skewcongruence, hence,itiskerneloftheprojection ofA2 ontooneofitsfactors. Without loss of generality let θ = {((c ,d ),(c ,d )) | c = c }. This means 1 1 2 2 1 2 that, for any e ∈ A and any (c ,d ,e),(c ,d ,e) ∈ R, we have c = c . How- 1 1 2 2 1 2 ever, (a,b,a),(b,a,a) ∈ R, a contradiction. The same argument applies when θ = {((c ,d ),(c ,d )) |d = d }. 1 1 2 2 1 2 Thus,tol isthetotalrelation,andthereis(c,d) ∈Qsuchthat(c,d,a),(c,d,b) ∈ 3 Rwhichimplies{(c,d)}×A ⊆ R. CLAIM 2. Forany(c′,d′)∈ pr R,thetuple(c′,d′,a) ∈ R. 1,2 Take a term operation g(x,y,z) such that g(a,b,c) = c′ and g(a,b,ϕ−1(d)) = ϕ−1(d′). Such an operation exists whenever c 6= ϕ−1(d), because every pair of 6 elementsofAisatrace. Then a b c g(a,b,c) g b , a , d = g(b,a,d)         a a a a      c′   = ϕ(g(ϕ−1(b),ϕ−1(a),ϕ−1(d)   a  c′ c′ c′ = ϕ(g(a,b,ϕ−1(d) = ϕ(ϕ−1(d′)) = d′ ∈ R.       a a a       What is left is to show that there are c,d such that (c,d,a) ∈ R and c 6= ϕ−1(d). Suppose c = ϕ−1(d). If Sg(a,c),Sg(c,b) 6= A, the a,b are connected in H(A). LetSg(a,c) = A,andhsuchthat h(a,c) = b. SinceRissymmetric withrespect toanypermutationofcoordinates,{c}×A×{d} ⊆ R. Inparticular,(c,c,d) ∈ R. Then a c b h a , c = b ,       b d a asc= ϕ−1(d)anda = ϕ−1(b). Thetuple(b,b,a)isasrequired. Thus, (a,a,a) ∈ R whichmeansthatthereisatermoperation f(x,y,z)such thatf(a,a,b) =f(a,b,a) = f(b,a,a) = a. Sinceϕisanautomorphism, wealso getf(b,b,a) = f(b,a,b) = f(a,b,b) = b,i.e.f isamajorityoperation on{a,b}. ✷ Proof:[Theorem 1] Suppose for contradiction that G(A) is disconnected. Let B be a minimal subalgebra of A such that G(B) is disconnected. Since the graph ofeveryproper subalgebra ofBisconnected, Bis2-generated, say, B = Sg(a,b). Letθ beamaximalcongruence ofB. Clearly, if G(B/ ) is connected then G(B) is connected. Therefore, B/ is θ θ tolerance free and of type 3. Take c,d ∈ B; let c′ = cθ,d′ = dθ. If Sg(c,d) 6= B then c,d are connected by the assumption made. Otherwise Sg(c′,d′) = B/ . By θ Proposition 4, either c′,d′ are connected in H(B/ ) and hence in G(B/ ), or c′d′ θ θ is an edge in G(B/ ). In the former case c,d are connected because every proper θ subalgebra ofB/ givesrisetoaproper subalgebra ofB. Inthelattercasecdisan θ edgeofG(B). Thus,G(B)isconnected, acontradiction. ✷ 3.3 Adding thick edges Generally,anedge,orevenathickedgeisnotasubalgebra. However,weshowthat everyidempotentalgebraAomittingtype1hasareductA′suchthatA′alsoomits 7 type one, but every its edge of semilattice or majority type is a subalgebra of A′. Moreover,sometyperestrictionsarealsoobserved. WesaythatG(A)issemilattice (semilattice/majority)-connected if every two vertices in G(A) are connected by a path consisting ofsemilattice (semilattice andstrict majority) edges. Forshort we willabbreviate ittos-connected andsm-connected. Theorem5 Let A be an idempotent algebra omitting type 1, ab an edge of G(A) of semilattice or strict majority type, and R = (aθab ∪ bθab) the thick edge ab. ab LetalsoF denotesetoftermoperations ofApreserving R ab ab (1)A′ = (A,F )omitstype1. ab (2)Ifabissemilattice andG(A)iss-connected, thenG(A′)iss-connected. (3)IfabisstrictmajorityandG(A)issm-connected, thenG(A′)issm-connected. We prove Theorem 5 by induction on the ‘structure’ of the algebra. The base case of this induction is given by strictly simple algebras. Recall that a simple algebra whose proper subalgebras are all 1-element is said to be strictly simple. Weneedthedescription offiniteidempotentstrictlysimplealgebras givenin[23]. Let G be a permutation group acting on a set A. By R(G) we denote the set ofoperations onApreserving eachrelationoftheform{(a,g(a)) |a ∈ A}where g ∈ G,andF(G)denotes thesetofidempotentmembersofR(G). Let A = (A;+,K)beafinitedimensionalvectorspaceoverafinitefieldK, K T(A)the group oftranslations {x+a | a ∈ A}, and End Athe endomorphism K ring of A. Then one can consider A as a module over End A. This module is K K denoted by A. (EndKA) Finally, letF0 denotethesetofalloperations preserving therelation k X0 = {(a ,...,a ) ∈ Ak |a = 0foratleastonei, k 1 k i 1 ≤ i≤ k} where0issomefixedelementofA,andletF0 = ∞ F0. ω k=2 k Theorem6([23]) Afinitestrictly simpleidempoteTnt algebraAistermequivalent tooneofthefollowingalgebras: (a) (A,F(G)) for a permutation group G on A such that every nonidentity memberofGhasatmostonefixedpoint; (b)(A,Term ( A))forsomevectorspace AoverafinitefieldK; id (EndKA) K (c) (A,F(G) ∩F0) for some k (2 ≤ k ≤ ω), some element 0 ∈ A and some k permutation group G acting on A such that 0 is the unique fixed point of every nonidentity memberofG; (d)(A,F)where|A| = 2andF contains asemilatticeoperation; (e)atwo-elementalgebra withanemptysetofbasicoperations. 8 Itcanbeeasilyshown(seee.g.[15])thatincase(c)Ahasatermzero-multiplication operation, thatabinaryoperation hsuchthath(x,y) = 0wheneverx 6= y. Proof:[of Theorem 5.] Let ab be an edge of semilattice type and f is a term operation such that f/ is asemilattice operation on B′ = {aθab,bθab}. Wewill θab omitindexθ everywhereitdoesnotleadtoaconfusion. LetA′ = (A;F′)where ab F′ is the set of binary term operations g of A such that g/ on B′ is either a θab projection orequals f/ . Thesubalgebra ofAgenerated byasetB ⊆ Awillbe θab denotedbySg (B),whilethesubalgebraofA′ generatedbythesamesetwillbe old denotebySg (B). Ingeneral, Sg (B) ⊆ Sg (B). new new old Claim1. f canbechosentosatisfytheidentityf(x,f(x,y)) = f(x,y). For every x ∈ A, we consider the unary operation g (y) = f(x,y). There is a x natural number n such that gnx is an idempotent transformation of A. Let n be x x theleastcommonmultipleofthen ,x ∈Aand x h(x,y) = f(x,f(x,...f(x,y)...)). ntimes Sincegn(y)isanidempotentforany|x ∈ A{z,weha}veh(x,h(x,y)) = gn(gn(y)) = x x x gn(y) = h(x,y). Finally,asiseasilyseenhequalsf on{aθab,bθab}. x Weprovethat,foranyc,d ∈A,thegraphG(Sg (c,d))isconnected. More- new over, if for every subalgebra B of A, G(B) is s-connected, then this holds also for every subalgebra of A′. We proceed by induction on order ideals of Sub(A′). To prove the base case for induction, suppose that for c,d ∈ A′, the algebra Sg (c,d)isstrictlysimple. ByTheorem6,wehavetoconsiderfivecases. new CASE 1.A. Sg (c,d)isaset. new In this case, Sg(c,d) = {c,d} and f (x,y) = x. If Sg (c,d) 6= {c,d} then {c,d} old there exists a term operation g of A′ such that g(c,d) 6∈ {c,d}. As is easily seen, the operation g′(x,y) = g(f(x,y),f(y,x)) equals f on B′; hence, it belongs to F′. However, g′(c,d) = g(c,d) 6∈ {c,d}, a contradiction with the assumption made. Thus,Sg (c,d) = {c,d}. old Then there is a term operation g of A which is either an affine or majority or semilattice operation on{c,d}. Theoperation g′(x,y,z) = g(f(x,f(y,z)),f(y,f(z,x)),f(z,f(x,y))) inthefirsttwocasesorg′(x,y) = g(f(x,y),f(y,x)inthelattercasebelongtoF′ andisanaffineormajorityorsemilattice operation on{c,d}respectively. CASE 1.B. Sg (c,d) = {c,d}isa2-elementsemilattice. new 9 Thereisnothing toproveinthiscase. CASE 1.C. Sg (c,d)isamodule. new Theoperationf onSg (c,d)hastheformf(x,y)= px+(1−p)yandeitherp new or1−pisinvertible. Supposethatpisinvertible andpn = 1foracertainn. Then set f′(x,y) = f(f(...f(x,y)...,y),y). ntimes Sincef andf′areidempotent,f′(x,y) =xonSg (c,d)andf′(x,y) = f(x,y) | {z } new onB′. Then, as in Case 1.A we show that Sg (c,d) = Sg (c,d). Therefore, new old B = Sg (c,d) is a strictly simple algebra. If B is 2-element then we get one of old the previous cases. Otherwise, Beither has azero-multiplication operation hor it is of the form (B;F(H)) for a certain permutation group H. In the former case, h(f(x,y),f(y,x)) belongs to F′ and is a zero-multiplication operation on B. In thelattercase,Bhasanoperationwhichiseitherasemilatticeormajorityoperation on {c,d}. Arguing as above we get an operation of A′ which is semilattice or majorityon{c,d}respectively. CASE 1.D. Sg (c,d)hasazero-multiplication operation h. new Let0bethezero-element. Thenc,dareconnected byedgesc0andd0. CASE 1.E. Sg (c,d) is of the form (B;F(H)) for a certain permutation group new H. (NotethatthisalgebrahastheBooleantype.) If there is no automorphism ϕ in H such that ϕ(c) = d and ϕ(d) = c, then, by Proposition4,A′ hasatermoperationg whichisasemilatticeoperationon{c,d}. So,letussuppose thatthereisanautomorphism swappingcandd. If Sg (c,d) has no operation which is semilattice on {c,d} then we are old done. Otherwise, let g be a term operation of A semilattice on {c,d} and h a term operation of A′ majority on {c,d}. If one ofh(x,x,y),h(x,y,x),h(y,x,x) is a semilattice operation on B′ then we proceed as before. Otherwise, h is a projection on B′; without loss of generality let it be the first projection. Then h′(x,y) = f(h(x,y,y),h(y,y,x)) equals f on B′ and is a projection on {c,d}. Wecompletetheproofasbefore. Now,supposethattheclaimprovedforallpropersubalgebrasofC = Sg (c,d). new Weconsider twocases. CASE 1.1. ThereisamaximalcongruenceθofCsuchthatf/θiscommutativeon C/ . θ ByClaim1,f isasemilatticeoperation on{cθ,f(cθ,dθ)}and{f(cθ,dθ),dθ}. 10