Linear identities in graph algebras

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1 Comment.Math.Univ.Carolin. 50,1(2009) Linear identities in graph algebras Agata Pilitowska Abstract. Wefindthebasisofalllinearidentitieswhicharetrueinthevarietyofentropic graph algebras. We apply it to describe the lattice of all subvarieties of power entropic graph algebras. Keywords: graph algebra, linear identity, entropic algebra, equational basis, lattice of subvarieties, power algebra of subsets Classification: Primary 08B15, 03C05, 03C13, 08C10; Secondary 17D99, 08A05, 08A40, 08A62, 08A99 1. Introduction In 1979 Caroline Shallon introduced in her dissertation[9] algebras associated withgraphs.let G=(V, E)bea(undirected)graphwithaset V ofverticesand aset E V V ofedges. Itsgraphalgebra A(G)=(V {0}, )isagroupoid with the multiplication defined as follows: { x, if (x, y) E, x y:= 0, otherwise. C. Shallon proved that many finite graph algebras are nonfinitely based. Moreoveritwasshownin[4]thatagraphalgebra A(G)isfinitelybasedifandonlyif it is entropic. Inthispaperwefindalllinearidentitieswhicharetrueinthevarietyofentropic graph algebras. Some new linear identity(3.1) true in this variety is crucial for the final result. Linear identities play an important rôle in the theory of power algebras. Thepower(complexorglobal)algebraCmA=(P(A), F)ofanalgebra(A, F) isthefamily P(A)ofallnon-emptysubsetsof Awithcomplexoperationsgiven by f(a 1,...,A n ):= {f(a 1,..., a n ) a i A i }, where A 1,..., A n Aand f Fisan n-aryoperation. While working on this paper, the author was partially supported by the INTAS grant# and Statutory Grant of Warsaw University of Technology# 504G

2 12 A. Pilitowska Power algebras were studied by several authors, for instance by G. Grätzer andh.lakser[5],s.whitney[6],a.shafaat[8],c.brink[3],i.bošnjakand R. Madarász[2]. Complexoperationsmaypreservesomeofpropertiesof(A, F), butnotall identitiestruein(a, F)mustbesatisfiedinCmA. Forexample, thepower algebraofagroupisnotagainagroup[5].foranarbitraryvariety V,G.Grätzer and H. Lakser determined the identities satisfied by the variety generated by CmA,forA V.Theyappliedtheirresulttodescribeallsubvarietiesofpower algebras of lattices and groups[5]. They showed that there is exactly one nontrivial variety of power algebras of lattices and there are exactly three non-trivial varieties of power algebras of groups. Inthispaperweshowthatthelatticeofallsubvarietiesofthevarietygenerated by power algebras of entropic graph algebras has eight elements. In Section 2 we recall main known results concerning graph algebras. Section 3 is devoted to linear identities satisfied in entropic graph algebras. In the last Section 4 we present some theorems concerning identities satisfied by power algebras of sets and describe all subvarieties of power entropic graph algebras. Wesaythatanalgebra(A, F)isidempotent ifeachelement a Aformsa one-elementsubalgebraof(a, F).Analgebra(A, F)isentropic,ifeachoperation f Fasamappingfromadirectpowerofthealgebraintothealgebraisactually a homomorphism. Wecallaterm tlinear,ifeveryvariableoccursin tatmostonce.anidentity t uiscalledlinear,ifbothterms tand uarelinear.anidentity t uiscalled regular,if tand ucontainthesamevariables.thenotation t(x 1,..., x n )means thattheterm tcontainsnoothervariablesthan x 1,..., x n (butnotnecessarily all of them). In the case of groupoid terms we will use non-brackets notation, as follows: { (...((xx1 )x 2 )...)x n, n 1, xx 1 x 2... x n := x, n=0. 2. Graph algebras Foreachnaturalnumber n N,let P n denote n-vertexgraphsintheformof apathwithoutloops,and L n denote n-vertexgraphsintheformofapathwith loops, as in the diagrams: P n : 1 L n : n Forgraphs Gand H, G+Hwilldenotethedisjointunionof Gand H,with noedgesbetween Gand H.... n

3 Linear identities in graph algebras 13 Itisnotdifficulttoseethatthedirectproductoftwographalgebrasisnot necessarilyagraphalgebra. Sotheclassofallgraphalgebrasdoesnotforma variety.letusdenoteby V A(G) thevarietygeneratedbyagraphalgebra A(G). Aswasshownin[1],thevarietyofallentropicgraphalgebrasisgeneratedby thealgebra A(P 2 + L 2 ).Thefollowingidentities (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) xy xyy x(yz) xy(yz) xyz xzy xy x(yx) x(yzu) x(yz)(yu) x(y(zu)) x(yz)(uz) x(yz)(uv) x(yv)(uz) x(xy) x(yy) xx(yz) x(yy)(zz) formabasisfor V A(P2 +L 2 ).Itiseasytoseethattheentropiclaw (E) xy(zw) xz(yw). follows by the above identities: xy(zw) (2.4) x(yx)(zw) (2.7) x(yw)(zx) (2.3) x(zx)(yw) (2.4) xz(yw). The lattice of all subvarieties of the variety of entropic graph algebras was discussedin[1]anditisgiveninfigure1. T denotes the trivial variety, SL is the variety of semilattices, LZ is the variety ofleftzerobands(groupoidsdeterminedbyidentity xy x), LNisthevarietyof left normal bands idempotent semigroups satisfying the additional left-normal law(2.3), U 1 isthevarietydefinedbytwoidentities:(2.1)and (2.10) xx xy and U 2 isthevarietydefinedbyallidentities(2.1) (2.9)andadditionalone (2.11) xx x(yy).

4 14 A. Pilitowska V A(P2 +L 2 ) U 2 V A(P3 +L 1 ) V V A(P1 +L 2 ) A(P2 +L 1 ) V A(P3 ) V A(P2 ) U 1 V A(P1 +L 1 ) LN V A(P1 ) LZ SL T Figure 1 The remaining subvarieties of entropic graph algebras have the following basis: V A(P1 ) : xy uz; V A(P1 +L 1 ) : x(yz) (xy)z, xy yx, xy x(yy); V A(P1 +L 2 ) : (2.1) (2.9), x(yz) (xy)z; V A(P2 ) : (2.1) (2.9), xx yy, x(yz) z(yx); V A(P2 +L 1 ) : (2.1) (2.9), x(yz) z(yx); V A(P3 ) : (2.1) (2.9), xx yy; V A(P3 +L 1 ) : (2.1) (2.9), x(yy) y(xx). As it was proved in[1] using the regular identities(2.1) (2.9), every term in thevariety V A(P2 +L 2 ) maybeexpressedinoneofthestandardforms: (i) x, (ii) x 1 (x 1 x 1 )(x 2 x 2 )...(x n x n ), for n 1, (iii) x 1 (y 1 x 1 )(y 2 x 2 )...(y n x n ) (x 1 y 1 )(y 2 x 2 )...(y n x n ), for n 1,where {x 1,..., x n } {y 1,..., y n }=. Thevariables x 1,..., x n will bereferredtoasbottomand y 1,..., y n willbereferredtoastopvariables.

5 Linear identities in graph algebras 15 Theorem 2.1([1]). An identity p q is derivable from identities(2.1) (2.9) if andonlyifbothsidesoftheidentity p qhavethesamestandardform,the sameleftmostvariablesand(inthecaseoftype(iii))thesametopandbottom variables. 3. Linear identities in entropic graph algebras Inthissectionwewillfindalllinearidentitieswhicharetrueinthevariety V A(P2 +L 2 ). Firstweprovethatallentropicgraphalgebrassatisfysomelinear identity which plays an important rôle in this paper. Theorem 3.1. The following linear identity (3.1) x(y(zt)) xy(tz) issatisfiedinthevariety V A(P2 +L 2 ). Proof: x(y(zt)) (2.6) x(yz)(tz) (2.2) xy(yz)(tz) (E) xyt(yzz) (2.1) xyt(yz) (E) xyy(tz) (2.1) xy(tz). LetΣbethefollowingsetofidentities: (E) xy(zt) xz(yt), Σ: (2.3) xyz xzy, (3.1) x(y(zt)) xy(tz). Theseidentitiesarealllinearandregularandtheyholdinanyvarietyofentropic graph algebras. Moreover, the linear identity(3.1) also follows from Σ: x(yz)(tw) (E) xt(yzw) (2.3) xt(ywz) (E) x(yw)(tz). Wewillshowthatanylinearidentitytruein V A(P2 +L 2 ) isaconsequenceofσ. Before proving this fact we present a sequence of technical lemmas. Let n 1and πbeanypermutationoftheset {1,2,..., n}.byusingidentities from Σ repeatedly we obtain: (3.2) (3.3) (3.4) (xy 1 y 2... y n )(zt 1 t 2... t n ) xz(y 1 t 1 )...(y n t n )(by(e)), xy 1 y 2...y n xy π(1) y π(2)...y π(n) (by(2.3)), x(y(z 1 t 1 )(z 2 t 2 )...(z n t n )) xy(t 1 z 1 )(t 2 z 2 )...(t n z n )(by(3.1)).

6 16 A. Pilitowska Moreover, as it was proved in[1], the following identity follows by entropicity and identities(2.3) and(2.7): (3.5) xy 1...y k (y k+1 z 1 )...(y k+n z n ) xy π(1)... y π(k) (y π(k+1) z δ(1) )...(y π(k+n) z δ(n) ), where πisapermutationontheset {1,2,..., k+ n}and δisapermutationon theset {1,2,..., n}. Lemma 3.2. The following identities (3.6) (3.7) xy 1... y n x(y 1 x)(y 2 x)...(y n x), x(yz 1 z 2... z n ) x(yx)(yz 1 )(yz 2 )...(yz n ) holdinthevariety V A(P2 +L 2 ) forany n 1. Proof:Theproofof(3.6)goesbyinduction.For n=1,theidentity(3.6)follows by(2.4). Nowletusassumethat(3.6)istrueforsome n >1. Thenusing(2.3) and(2.4) we obtain xy 1... y n+1 x(y 1 x)(y 2 x)...(y n x)y n+1 (2.3) xy n+1 (y 1 x)(y 2 x)...(y n x) (2.4) x(y n+1 x)(y 1 x)(y 2 x)...(y n x) (2.3) x(y 1 x)(y 2 x)...(y n+1 x). The identity(3.7) follows directly by(3.6) and(3.4): x(yz 1 z 2... z n ) (3.6) x(y(z 1 y)(z 2 y)...(z n y)) (3.4) xy(yz 1 )...(yz n ) (2.4) x(yx)(yz 1 )...(yz n ). Lemma3.3. Let n, k 0.ThefollowingidentitiesfollowfromthesetΣ: (3.8) (3.9) (3.10) (3.11) x(y(t 1 t 2 z 1... z n )) xyz 1... z n (t 2 t 1 ), (x(y 1 y 2... y n ))(t 1 t 2 z 1... z k ) (x(y 1 y 2...y n z 1... z k ))(t 1 t 2 ), (x(ty 1... y n w))z (x(ty 1... y n ))(zw), (x(z 1 y 1... y n ))(z 2 w) (x(z 2 y 1... y n ))(z 1 w),

7 Linear identities in graph algebras 17 (3.12) (yy 1 y 2... y n )(zz 1 z 2... z k ) { (y(zz1... z k n ))(y 1 z k n+1 )...(y n z k ), if k > n 0 yy 1... y n k z(y n k+1 z 1 )...(y n z k ), if n k 0 (3.13) (x(z 1 y 1...y k ))(z 2 y k+1 )...(z n+1 y k+n ) (x(z π(1) y δ(1)... y δ(k) ))(z π(2) y δ(k+1) )...(z π(n+1) y δ(k+n) ), foranypermutation δontheset {1,2,..., k+ n}andanypermutation πonthe set {1,2,..., n+1}. Proof:1.For n=0,theidentity(3.8)followsby(3.1).if n >0,then x(y(t 1 t 2 z 1... z n 1 z n )) (3.1) (xy)(z n (t 1 t 2 z 1... z n 1 )). Byinductionon nand(3.3)weobtain (xy)(z n (t 1 t 2 z 1... z n 1 )) xyz n z 1... z n 1 (t 2 t 1 ) (3.3) xyz 1... z n (t 2 t 1 ). 2.For n=0and k=0,theidentity(3.9)isobvious.if n >0and k >0,then (x(y 1 y 2... y n ))(t 1 t 2 z 1... z k 1 z k ) (3.1) x(y 1 y 2...y n (z k (t 1 t 2 z 1...z k 1 ))) (3.3) x(y 1 (z k (t 1 t 2 z 1... z k 1 ))y 2... y n ). Using(3.8)weconcludethat y 1 (z k (t 1 t 2 z 1... z k 1 )) y 1 z k z 1...z k 1 (t 2 t 1 ).Thus x(y 1 (z k (t 1 t 2 z 1... z k 1 ))y 2... y n ) x(y 1 z k z 1... z k 1 (t 2 t 1 )y 2... y n ) (3.3) x(y 1 y 2...y n z 1...z k (t 2 t 1 )) (3.1) (x(y 1 y 2... y n z 1...z k ))(t 1 t 2 ). 3.For n=0,theidentity(3.10)followsby(2.3)andentropicity.if n >0,then (x(ty 1... y n w))z (2.3) (xz)(ty 1... y n w) (E) (x(ty 1... y n ))(zw). 4.For n=0,theidentity(3.11)followsbyentropicity.if n >0,then (x(z 1 y 1...y n ))(z 2 w) (2.3) (x(z 2 w))(z 1 y 1... y n ) (E) (x(z 1 y 1...y n 1 ))(z 2 wy n ). Byinductionon n,(2.3)and(2.7)weobtain (x(z 1 y 1... y n 1 ))(z 2 wy n ) (x(z 2 wy 1... y n 1 ))(z 1 y n ) (2.3) (x(z 2 y 1...y n 1 w))(z 1 y n ) (2.7) (x(z 2 y 1...y n ))(z 1 w). 5. The identity(3.12) follows directly by(3.2). 6. The identity(3.13) follows by(2.3),(2.7) and(3.11). Thenexttheoremdescribesalllineartermsinthevariety V A(P2 +L 2 ).

8 18 A. Pilitowska Theorem3.4. Everytermin V A(P2 +L 2 ) maybeexpressedinoneofthefollowing standard forms: (I1) (I2) xy 1...y k (t 1 w 1 )...(t n w n ), k 0, n 0; (x(y 1 y 2... y k ))(t 1 w 1 )...(t n w n ), k 2, n 0. Proof: Weshow(usingΣ)thatthesetoftermsoftheform(I1)and(I2)is closed under multiplication. (a)let γ xy 1...y k (t 1 w 1 )...(t n w n )and δ ab 1... b l (c 1 d 1 )...(c m d m ). By (3.3)and(3.4)wehave γδ [xy 1... y k (t 1 w 1 )...(t n w n )][ab 1... b l (c 1 d 1 )...(c m d m )] If l > k 0,then (3.3) xy 1... y k (ab 1...b l (c 1 d 1 )...(c m d m ))(t 1 w 1 )...(t n w n ) (3.4) ((xy 1... y k )(ab 1... b l ))(d 1 c 1 )...(d m c m )(t 1 w 1 )...(t n w n ). ((xy 1... y k )(ab 1... b l ))(d 1 c 1 )...(d m c m )(t 1 w 1 )...(t n w n ) (3.12) (x(ab 1... b l k ))(y 1 b l k+1 )...(y k b l )(d 1 c 1 )...(d m c m )(t 1 w 1 )...(t n w n ). Hence γδcanbewrittenintheform(i2). For k l 0, ((xy 1...y k )(ab 1... b l ))(d 1 c 1 )...(d m c m )(t 1 w 1 )...(t n w n ) (3.12) xy 1... y k l a(y k l+1 b 1 )...(y k b l )(d 1 c 1 )...(d m c m )(t 1 w 1 )...(t n w n ), and γδhastheform(i1). (b)let γ xy 1... y k (t 1 w 1 )...(t n w n )and δ (a(b 1 b 2... b l ))(c 1 d 1 )...(c m d m ) for l 2.Then γδ [xy 1... y k (t 1 w 1 )...(t n w n )][(a(b 1 b 2... b l ))(c 1 d 1 )...(c m d m )] (3.3) x((a(b 1 b 2...b l ))(c 1 d 1 )...(c m d m ))y 1...y k (t 1 w 1 )...(t n w n ) (3.4),(3.3) x(a(b 1 b 2... b l ))y 1... y k (d 1 c 1 )...(d m c m )(t 1 w 1 )...(t n w n ). By(3.8)wehavethat x(a(b 1 b 2... b l )) xab 3... b l (b 2 b 1 ).Consequently x(a(b 1 b 2...b l ))y 1...y k (d 1 c 1 )...(d m c m )(t 1 w 1 )...(t n w n ) xab 3... b l y 1... y k (b 2 b 1 )(d 1 c 1 )...(d m c m )(t 1 w 1 )...(t n w n ),

9 Linear identities in graph algebras 19 and γδisreducedtotheform(i1). (c)let γ (a(b 1 b 2...b l ))(c 1 d 1 )...(c m d m )and δ xy 1... y k (t 1 w 1 )...(t n w n ), for l 2.Then γδ [(a(b 1 b 2... b l ))(c 1 d 1 )...(c m d m )][xy 1... y k (t 1 w 1 )...(t n w n )] (3.3) (a(b 1 b 2... b l ))(xy 1...y k (t 1 w 1 )...(t n w n ))(c 1 d 1 )...(c m d m ) (3.4) (a(b 1 b 2... b l ))(xy 1...y k )(w 1 t 1 )...(w n t n )(c 1 d 1 )...(c m d m ). If k=0,thenby(3.10) γδ (a(b 1 b 2...b l 1 ))(xb l )(w 1 t 1 )...(w n t n )(c 1 d 1 )...(c m d m ). If k 1,thenby(3.9)wehave It follows that (a(b 1 b 2... b l ))(xy 1... y k ) (3.9) (a(b 1 b 2... b l y 2...y k ))(xy 1 ). (a(b 1 b 2... b l ))(xy 1... y k )(w 1 t 1 )...(w n t n )(c 1 d 1 )...(c m d m ) (a(b 1 b 2... b l y 2...y k ))(xy 1 )(w 1 t 1 )...(w n t n )(c 1 d 1 )...(c m d m ). Hence,forany k 0, γδmaybereducedtotheform(i2). (d) Let γ (x(y 1 y 2... y k ))(t 1 w 1 )...(t n w n ) and δ (a(b 1 b 2... b l ))(c 1 d 1 )... (c m d m ),for k, l 2.By(3.4)and(E),wehave γδ [(x(y 1 y 2... y k ))(t 1 w 1 )...(t n w n )][(a(b 1 b 2...b l ))(c 1 d 1 )...(c m d m )] (3.4) (x((y 1...y k )(w 1 t 1 )...(w n t n )))(a((b 1 b 2...b l )(d 1 c 1 )...(d m c m ))) (E) xa(((y 1... y k )(w 1 t 1 )...(w n t n ))((b 1 b 2...b l )(d 1 c 1 )...(d m c m ))). Thusby(a)and(b),alsointhiscase, γδreducestotherequiredform. Proposition3.5. ThesetΣisabasisforalllinearidentitiestrueinthevariety of entropic graph algebras. Proof:Accordingtoresultsof[1],eachidentitywhichissatisfiedin V A(P2 +L 2 ) isregular. Let p q be linearandregularidentitytruein V A(P2 +L 2 ). By Theorem3.4,theterms pand qareinoneofthetwostandardforms. Case 1. Let pand qbeoftype(i1). Therearedistinctvariables {x, y 1,..., y k, y k+1..., y k+n, z 1,..., z n }={a, b 1,..., b l, b l+1,..., b l+m, c 1,..., c m },such that p xy 1...y k (y k+1 z 1 )...(y k+n z n )

10 20 A. Pilitowska and By(3.6), and q ab 1...b l (b l+1 c 1 )...(b l+m c m ). p x(y 1 x)...(y k x)(y k+1 z 1 )...(y k+n z n ) q a(b 1 a)...(b l a)(b l+1 c 1 )...(b l+m c m ). ByTheorem2.1,theidentity p qissatisfiedin V A(P2 +L 2 ) ifandonlyif x=a, {x, z 1,..., z n } = {a, c 1,..., c m }and {y 1,..., y k+n } = {b 1,..., b l+m }. This impliesthat n=m, k= landtherearepermutations πon {1,2,..., k+ n}and δon {1,2,..., n}suchthat q xy π(1)... y π(k) (y π(k+1) z δ(1) )...(y π(k+n) z δ(n) ). Consequently, by identity(3.5), the identity p q follows from Σ. Case2.Let pand qbeoftype(i2).therearedistinctvariables {x, z 1,..., z n+1, y 1,..., y k+n }={a, b 1,..., b l, b m+1, c 1,..., c l+m },suchthat and k, l 2. By the identity(3.7), p (x(z 1 y 1... y k ))(z 2 y k+1 )...(z n+1 y k+n ), q (a(b 1 c 1... c l ))(b 2 c l+1 )...(b m+1 c l+m ), p x(z 1 x)(z 1 y 1 )...(z 1 y k )(z 2 y k+1 )...(z n+1 y k+n ) and q a(b 1 a)(b 1 c 1 )...(b 1 c l )(b 2 c l+1 )...(b m+1 c l+m ). ByTheorem2.1,theidentity p qissatisfiedin V A(P2 +L 2 ) ifandonlyif x= a, {x, y 1,..., y k+n }={a, c 1,..., c l+m }and {z 1,..., z n+1 }={b 1,..., b m+1 }. Thisimpliesthat n=m, k= landtherearepermutations δon {1,2,..., k+ n} and πon {1,2,..., n+1}suchthat q (x(z π(1) y δ(1)...y δ(k) ))(z π(2) y δ(k+1) )...(z π(n+1) y δ(k+n) ). Hence,byidentity(3.13),theidentity p qfollowsfromσ. Case3.Let pbeoftype(i1)and qbeoftype(i2).asbefore, p xy 1...y k (y k+1 z 1 )...(y k+n z n )

11 Linear identities in graph algebras 21 and q (a(b 1 c 1... c l ))(b 2 c l+1 )...(b m+1 c l+m ), where {x, z 1,..., z n, y 1,..., y k+n }={a, b 1,..., b m+1, c 1,..., c l+m }and l 2. SimilarlyasinCase1andCase2,theidentities(3.6),(3.7)andTheorem2.1 implythattheidentity p q issatisfiedin V A(P2 +L 2 ) ifandonlyif x=a, {x, z 1,..., z n } = {a, c 1,..., c l+m }and {y 1,..., y k+n } = {b 1,..., b m+1 }. It followsthat n=m+land m+1=k+n.hence k+l=1.butweassumedthat l 2,sothisgivesacontradiction. Thisshowsthatanylinearidentityholdingin V A(P2 +L 2 ) isaconsequenceofσ. 4. The variety of power entropic graph algebras Usingtheresultoftheprevioussectionwewilldescribethelatticeofallsubvarieties of power entropic graph algebras. Let Vbeavariety. WewilldenotebyCmVthevarietygeneratedbypower algebras of algebras in V, i.e., CmV:=HSP({CmA A V}). Evidently, V CmV,becauseeveryalgebraAin V embedsintocmaby x {x}.g.grätzerandh.lakserprovedthefollowingtheorem. Theorem 4.1([5]). Let V be a variety. The variety CmV satisfies precisely those identities resulting through identification of variables from the linear identities truein V. Forexample,anidempotentlawissatisfiedinthevarietyCmVifandonlyif itisaconsequenceoflinearidentitiestruein V. As an immediate corollary of the Theorem 4.1 we have the following result. Corollary4.2([5]). Foravariety V,CmV= Vifandonlyif Visdefinedbya set of linear identities. AsitwasshowninSection3,anylinearidentitytruein V A(P2 +L 2 ) isderivable fromthesetσ.hence,bytheorem4.1,weobtainthefollowingresult. Proposition4.3. CmV A(P2 +L 2 ) =Mod(Σ). Similarlyasinthecaseofthevariety V A(P2 +L 2 ) wecanshowthatvarieties CmV A(P3 ),CmU 2andCmV A(P3 +L 1 ) arealsodefinedbythesetσ. Now we will give a description of varieties generated by power algebras from remainingsubvarietiesof V A(P2 +L 2 ).Thevarieties T, LZand V A(P 1 ) aredefined onlybylinearidentities.hencebycorollary4.2,cmt = T,CmLZ= LZand CmV A(P1 ) = V A(P 1 ).

12 22 A. Pilitowska Thevarieties SLand V A(P1 +L 1 ) satisfyassociativityandcommutativity. Hence,CmSL=CmV A(P1 +L 1 ) isthevariety CSofcommutativesemigroups. Itiseasytoseethatbyleft-normallaw(2.3)andassociativity,alinearregular identity p qistrueinthevariety LNor V A(P1 +L 2 ) ifandonlyifterms pand qhavethesameleftmostvariable.thisimpliesthatcmln=cmv A(P1 +L 2 ) is the variety LS of left-normal semigroups. Moreover,thevariety U 1 satisfiestheassociativelaw. Thisimpliesthateach linearidentityin U 1 isaconsequenceoftheidentity xy xzandassociativity. HenceCmU 1 coincideswiththevariety USdefinedby xy xzandassociativity. Now let us consider the following linear identity: (4.1) x(yz) z(yx). Itisnotdifficulttoseethattheidentity(3.1)isaconsequenceofentropicity, left-normality and the identity(4.1): x(y(zt)) (4.1) zt(yx) (2.3) z(yx)t (4.1) x(yz)t (2.3) xt(yz) (E) xy(tz). LetΓbethefollowingsetofidentities: Γ: Lemma4.4. For n 1,theidentity (E) xy(zt) xz(yt), (2.3) xyz xzy, (4.1) x(yz) z(yx). x 1 y 1 (y 2 x 2 )...(y n x n ) x π(1) y δ(1) (y δ(2) x π(2) )...(y δ(n) x π(n) ) followsfromthesetγforanypermutations πand δoftheset {1,2,..., n}. Proof: This follows directly by(3.5) and(4.1). Theorem4.5. Anylinearidentityholdinginthevariety V A(P2 +L 1 ) isaconsequenceofthesetγ. Proof:UsingthesamemethodsasinproofsofTheorem3.4andProposition3.5 weobtainthateachlinearregularidentitytruein V A(P2 +L 1 ) isoneofthetwo following types: x 1 y 1... y k (y k+1 x 2 )...(y n+k x n+1 ) x δ(1) y π(1)...y π(k) (y π(k+1) x δ(2) )...(y π(n+k) x δ(n+1) ),

13 Linear identities in graph algebras 23 foranypermutation πontheset {1,..., n+k}andanypermutation δonthe set {1,..., n+1},or (x 1 (y 1 x 2...x k ))(y 2 x k+1 )...(y n+1 x k+n ) (x δ(1) (y π(1) x δ(2)...x δ(k) ))(y π(2) x δ(k+1) )...(y π(n+1) x δ(k+n) ), foranypermutation πontheset {1,..., n+1}andanypermutation δonthe set {1,..., n+k}. By identities(3.5),(3.13),(4.1) and(4.2), both follow from Γ. Similarly,wecanshowthatanylinearidentitytruein V A(P2 ) isalsoaconsequence of Γ. All these observations prove the following proposition. Proposition 4.6. There are eight varieties of power entropic graph algebras: CmV A(P2 +L 2 ) =CmV A(P 3 ) =CmU 2=CmV A(P3 +L 1 ) =Mod(Σ) CmV A(P2 +L 1 ) =CmV A(P 2 ) =Mod(Γ) CmLN=CmV A(P1 +L 2 ) = LS=Mod(xyz xzy, xyz x(yz)) CmSL=CmV A(P1 +L 1 ) = CS=Mod(xy yx, xyz x(yz)) CmU 1 = US=Mod(xy xz, xyz x(yz)) CmV A(P1 ) = V A(P 1 ) =Mod(xy uz) CmLZ= LZ=Mod(xy x) CmT = T =Mod(x y) ThelatticeofallsubvarietiesofCmV A(P2 +L 2 ) isgiveninfigure2: Mod(Σ) Mod(Γ) LS CS US V A(P1 ) LZ T Figure 2

14 24 A. Pilitowska References [1] Baker K.A., McNulty G.F., Werner H., The finitely based varieties of graph algebras, Acta Sci. Math.(Szeged) 51(1987), [2] Bošnjak I., Madarász R., On power structures, Algebra Discrete Math. 2(2003), [3] Brink C., Power structures, Algebra Universalis 30(1993), [4] Davey B.A., Idziak P.H., Lampe W.A., McNulty G.F., Dualizability and graph algebras, Discrete Math. 214(2000), [5] Grätzer G., Lakser H., Identities for globals(complex algebras) of algebras, Colloq. Math. 56(1988), [6] Grätzer G., Whitney S., Infinitary varieties of structures closed under the formation of complex structures, Colloq. Math. 48(1984), [7] McNulty G.F., Shallon C., Inherently nonfinitely based finite algebras, R. Freese, O. Garcia (Eds.), Universal Algebra and Lattice Theory(Puebla, 1982), Lecture Notes in Mathematics, 1004, Springer, Berlin, 1983, pp [8] Shafaat A., On varieties closed under the construction of power algebras, Bull. Austral. Math. Soc. 11(1974), [9] Shallon C.R., Nonfinitely based binary algebras derived from lattices, Ph.D. Thesis, University of California at Los Angeles, Faculty of Mathematics and Information Science, Warsaw University of Technology, Plac Politechniki 1, Warsaw, Poland apili@mini.pw.edu.pl URL: apili (Received May 23, 2008)

THE LATTICE OF SUBVARIETIES OF SEMILATTICE ORDERED ALGEBRAS

THE LATTICE OF SUBVARIETIES OF SEMILATTICE ORDERED ALGEBRAS A. PILITOWSKA 1 AND A. ZAMOJSKA-DZIENIO 2 Abstract. This paper is devoted to the semilattice ordered V-algebras of the form (A, Ω, +), where