LLoop conditions
Miroslav Olˇs´akNovember 15, 2018
Abstract
We prove that the existence of a term s satisfying s ( x, y, y, z, z, x ) = s ( y, x, z, y, x, z ) is the weakest non-trivial strong Maltsev condition givenby a single identity. Acknowledgement
This research was supported by the Czech Science Foundation (GA ˇCR), grant13-01832S.
One of the most interesting by-products of the research on the fixed-templateconstraint satisfaction problems is the result of Mark Siggers [11], which saysthat there exists a weakest non-trivial strong Maltsev condition for idempotentlocally finite varieties.Let us recall that a strong Maltsev condition (see [5, 9, 4]) is a conditionfor a variety (or an algebra) postulating the existence of finitely many termssatisfying a given finite set of identities. Such conditions can be compared bytheir strength: a condition C is weaker than D if each variety satisfying D also satisfies C . The weakest conditions are those satisfied in every variety,we call them trivial . The concept of strength can be naturally relativized tospecial types of varieties, such as idempotent locally finite varieties in the abovementioned result of Siggers.His weakest non-trivial condition has an especially simple form: it is givenby a single linear identity in one operation symbol appearing on both sides,namely s ( x, y, y, z, z, x ) = s ( y, x, z, y, x, z ) . A well known fact, stated here as Proposition 1, is that strong Maltsev con-ditions of this form can be characterized by the existence of a loop in certainbinary relations compatible with algebras in the variety. This structural prop-erty of compatible relations proved useful (see [1, 2]) and inspired the followingterminology.
Definition 1. A loop condition is a strong Maltsev condition given by a singleidentity of the form t ( x , . . . , x n ) = t ( y , . . . , y n ) , where x , . . . , x n , y , . . . , y n are variables. a r X i v : . [ m a t h . L O ] F e b n idempotent locally finite varieties, the 6-ary loop condition by Siggers isequivalent to many other loop conditions, such as the existence of a 4-ary term t satisfying t ( r, a, r, e ) = t ( a, r, e, a ) [7].What happens when we drop the local finiteness assumption? While thereis no weakest non-trivial strong Maltsev condition for general varieties [12, 8],it turned out that there is one for idempotent varieties [10]; for instance, the exis-tence of a 6-ary t satisfying t ( x, y, y, y, x, x ) = t ( y, x, y, x, y, x ) = t ( y, y, x, x, x, y ).Such a condition cannot be a loop condition: A. Kazda [6] proved that thefree idempotent algebra over { x, y } in the signature consisting of one ternaryoperation symbol w modulo the weak near unanimity identities w ( x, y, y ) = w ( y, x, y ) = w ( y, y, x ) does not satisfy any non-trivial loop condition.On the other hand, a consequence of the main result of this paper, Theo-rem 1, is that the existence of a 6-ary Siggers term is a weakest non-trivial loopcondition in general , that is, for varieties that are not necessarily idempotentor locally finite. To describe the main result, it will be convenient to assign adirected graph (digraph) to each loop condition in a natural way: Definition 2.
Let C be a loop condition given by an identity t ( x , . . . , x n ) = t ( y , . . . , y n ) , where x , . . . , x n , y , . . . , y n are variables from a set V . The di-graph of C , denoted G C , is the digraph ( V, E ) with vertex set V equals to thevariable set and edge set E = { ( x i , y i ) : i = 1 , . . . , n } . For example, the digraph of the loop condition s ( x, y ) = s ( y, x ) is K andthe digraph of the 6-ary loop condition by Siggers is K , where K i denotes thecomplete loopless digraph on i vertices.It is easy to see that two loop conditions, whose associated digraphs areisomorphic, are equivalent. Therefore, we may talk about “loop condition G ”(or sometimes “ G loop condition”) instead of “a loop condition whose associateddigraph is G ”. Also note that G C contains a loop if and only if C is trivial, i.e.,satisfied in every algebra.Our main result, Theorem 1, fully classifies the stregth of undirected loopconditions: Each such nontrivial condition is either equivalent to the existenceof a commutative term (condition K ) or the existence of a Siggers 6-ary term(condition K ). xy z Figure 1: The digraph of the Siggers term. An n -ary operation f on a set A is compatible with an m -ary relation R ⊆ A m ,or R is compatible with f , if f ( r , . . . , r n ) ∈ R for any r , . . . , r n ∈ R . Here(and later as well) we abuse the notation and use f also for the n -ary operationon A m defined from f coordinate-wise.2n algebra A = ( A, f , f , . . . ) is said to be compatible with a relationalstructure A = ( A, R , R , . . . ) if all the operations f , f , . . . are compatiblewith all the relations R , R , . . . . Digraph is a relational structure G = ( G, E ) with one binary relation E . If E is symmetric, then the digraph G is called undirected . A loop in G is a pairof the form ( x, x ) ∈ E .We will extensively use a standard method for building compatible relationsfrom existing ones – primitive positive ( pp , for short) definitions. A relation R is pp-definable from relations R , . . . , R n if it can be defined by a first orderformula using variables, existential quantifiers, conjunctions, the equality rela-tions, and predicates R , . . . , R n . Clauses in pp-definitions are also referred toas constraints . Recall that R , . . . , R n are compatible with an algebra, then sois R .It is sometimes helpful to visualize a pp-definition of k -ary relation R from adigraph G as a digraph H with k distinguished vertices u , . . . , u k : the verticesof H are the variables, its edges correspond to the constraints, and the distin-guished vertices are the free variables. Observe that ( v , . . . , v k ) is in R if andonly if there is a digraph homomorphism H → G which maps u i to v i for each i = 1 , . . . , k . In a similar way, we can visualize pp-definitions from a relationalstructure consisting of more than one binary relation.A pp-power of a relational structure A on A is a relational structure B on A l whose relations are pp-definable from A in the sense that a k -ary relation from B , regarded as a ( k · l )-ary relation on A , is pp-definable from A . Observe thatif A is compatible with an algebra A , then the pp-power B is compatible withthe algebraic power A l .For more background on pp-definitions and its relevance for constraint sat-isfaction problems we refer the reader to [3].Finally, we state the promised correspondence between loop conditions andloops in digraphs. Proposition 1.
Let V be a variety and C a loop condition. The followingconditions are equivalent.(i) V satisfies C .(ii) For every A ∈ V and for every digraph G = ( A, G ) compatible with A ,the following holds: If there is a digraph homomorphism G C → G then G contain a loop.(iii) For every A ∈ V and for every digraph G = ( A, G ) compatible with A , thefollowing holds: If a subdigraph of G is isomorphic to G C then G containsa loop.Proof. (i) ⇒ (ii) Let C be of the form t ( x , . . . , x n ) = t ( y , . . . , y n ) and f : G C → G be a digraph homomorphism. Then ( f ( x i ) , f ( y i )) ∈ G for all i = 1 , . . . , n .By compatibility of G with A also( t ( f ( x ) , . . . , f ( x n )) , t ( f ( y ) , . . . , f ( y n ))) ∈ G, Since equality t ( f ( x ) , . . . , f ( x n )) = t ( f ( y ) , . . . , f ( y n ))is ensured by C , we get the desired loop in G .3ii) ⇒ (iii) Trivial.(iii) ⇒ (i) Let F be the free algebra in V generated by the vertices of G C andlet E be the subuniverse of F generated by the edges of G C . The set E is theedge–set a digraph compatible with F containing G C as a subgraph. Therefore,by (iii), there is loop ( a, a ) in E . The pair ( a, a ) is generated from edges of G C , so there is a term operation t of F taking edges of G C as arguments andreturning ( a, a ). Thus, in F , the term t satisfies the loop condition C , at leastwhen we plug in the generators. By the universality of free algebra, the term t satisfies C in general. Corollary 1.
Let C , D be loop conditions. If there is a digraph homomorphism G C → G D , then C implies D . In particular, C implies D whenever G C is asubdigraph of G D . Corollary 2.
Loop conditions with homomorphically equivalent digraphs areequivalent. In particular, loop conditions with isomorphic digraphs are equiva-lent.
In this section, we will focus on loop conditions with undirected digraphs, brieflygraphs.We start with several simple consequences of Corollary 1.
Proposition 2.
1. Any loop condition of a bipartite graph is equivalent tothe edge loop condition (commutativity).2. The ( l + 2) -cycle loop condition implies the l -cycle loop condition for anyodd length l ≥ .3. The n -clique loop condition implies the ( n + 1) -clique loop condition forany size n ≥ .4. For any non-bipartite graph G there is an odd length l ≥ such that the l -cycle loop condition implies the loop condition given by G .5. For any loopless digraph G there is a size n ≥ such that the loop conditiongiven by G implies the n -clique loop condition. Figure 2: Scheme of the easy implications between undirected loop conditionsgiven by Proposition 2. 4ur aim in the rest of the chapter is to reverse the second and the thirdimplication, that is, (2 n + 3)-cycle to (2 n + 1)-cycle and ( n + 2)-clique to ( n + 3)-clique, where n ≥
1. It will follow that all the loopless non-bipartite graphs areequivalent as loop conditions and that they are weakest among all non-trivialloop conditions.
Proposition 3.
The l -cycle loop condition implies the ( l + 2) -cycle loop condi-tion for any odd length l ≥ .Proof. There is a graph homomorphism from the l -cycle to the ( l + 2)-cyclebecause both cycles are odd and l ≥ l + 2. It is thus sufficient to show that the l -cycle loop condition implies the l -cycle one.We use Proposition 1. Let A be an algebra on a set A such that everydigraph compatible with A containing a homomorphic image of the l -cycle hasa loop. We need to prove that every compatible digraph G = ( A, G ) containinga homomorphic image of the l -cycle has a loop. Take such a digraph G wherethe cycle is formed by vertices v , v , . . . , v l − . We may assume that G issymmetric, since the set of symmetric edges of G is clearly pp-definable from G . G H
Figure 3: Getting a loop from 9-cycle using the triangle loop condition.We construct a binary relation H on A : vertices x and y are by definition H -adjacent if there is a G -walk from x to y of length l . Since H is pp-defined froma compatible relation, it is compatible with A too. Vertices v , v l , v l , . . . , v l ( l − form a cycle in H of length l . By the assumption on algebra A , there is a loopin H . A loop in H is a homomorphic image of an l -cycle in G . Using theassumption on A again we get a loop in G . Proposition 4.
The ( n + 1) -clique loop condition implies the n -clique loopcondition for any n ≥ .Proof. As in the previous proposition, we use Proposition 1. Let A on a set A be an algebra satisfying the ( n + 1)-clique loop condition. Let G = ( A, G ) be adigraph compatible with A containing an n -clique a , a , . . . , a n as a subgraph.It suffices to prove that G has to have a loop. As before we may assume that G is a graph.Let us pp-define a 4-ary relation R on A as follows. R ( u, v, x, y ) if andonly if there are elements x , . . . , x n − , w ∈ A such that all the vertices x i arepairwise G -adjacent to each other, they are also G -adjacent to vertices x, y, v, w and moreover G ( u, w ) , G ( w, x ) , G ( v, y ). From R we pp-define a binary relation F ( x, y ) ⇔ ∃ u ∈ A : R ( u, u, x, y ). 5 − n − R ( u, v, x, y ) F ( x, y ) xy xyvu w Figure 4: A vizualisation of the definitions of R and F .Observe that the pp-definition of F is symmetric, so F itself is symmetric.We regard F as the set of edges of a graph F = ( A, F ) compatible with A . Claim 1.
Let u, v, x, y ∈ { a , . . . , a n } . Then R ( u, v, x, y ) whenever one of thefollowing condition are met:(a) u (cid:54) = v and x = y (cid:54) = v .(b) u = v and x = u = v , y (cid:54) = x . To prove the claim, we need to find correct values of variables x , . . . , x n − , w in the definition of R to meet the constraints of R . We do it separately for thetwo cases.(a) We set w = v and variables x i to vertices a i different from v , x .(b) We set w = y and variables x i to vertices a i different from x , y .The next claim immediately follows from the case (b) of the previous one. Claim 2. If x, y ∈ { a , . . . , a n } and x (cid:54) = y , then F ( x, y ) . Finally we define a digraph Q = ( A , Q ), where the binary relation Q isdefined as follows: Q (( u , u ) , ( v , v )) if and only if there are x , x , . . . , x n +1 such that every pair of different indices i , i , with the possible exception ofpairs { , } and { , } , satisfies F ( x i , x i ) and, moreover, R ( u , v , x , x ) and R ( u , v , x , x ). Claim 3. If u , u , v , v ∈ { a , . . . , a n +1 } and ( u , u ) (cid:54) = ( v , v ) , then Q (( u , u ) , ( v , v )) . To prove the claim, let variables x , . . . , x n +1 be as in the definition of Q .We satisfy the constraints of Q by means of Claims 1, 2. Because of F clauses inthe definition of Q , all pairs of variables from { x , . . . , x n +1 } should differ, withtwo possible exceptions x = x and x = x . We analyse cases according toequalites among u , u , v , v . In each case we use Claim 1 and assign suitablevalues to x , x , x , x .(a) u (cid:54) = v and u (cid:54) = v : We want to use (a), that is, we want to have x = x (cid:54) = v and x = x (cid:54) = v . First we choose x = x , then x = x different from v and x . This is possible since n ≥ − Q (( u , u ) , ( v , v )) v u n − n − v u FGx x x x Figure 5: A visualization of the definitions of Q .(b) u = v and u (cid:54) = v : We want to satisfy x = u , y (cid:54) = x (and use (b)),and x = x (cid:54) = v (and use (a)). So we put x = u , then choose a valuefor x = x different from x , v and, finally, we choose a value for x different from x , x .(c) u (cid:54) = v and u = v is analogous to the previous case.(d) u = v and u = v can not happen since ( u , u ) (cid:54) = ( v , v ).In every case x = x or x = x , therefore the remaining variables x i can becompleted so that all the constraints of Q are satisfied.Let us finish the proof. The digraph Q is compatible with the algebra A since it is a pp-power of G . Moreover, Q contains a clique of size n ≥ n + 1.The n -clique loop condition holds also for A , so there is a loop in Q .The loop in Q is represented by elements x , . . . , x n +1 in A such that F ( x i , x j )whenever i (cid:54) = j . Since F is pp-defined from G , it is compatible with A . There-fore, there is a loop in F .Finally, a loop in F yields a ( n + 1)-clique in G and, consequently, the soughtafter loop in G . Theorem 1.
There are exactly three equivalence classes of loop conditions givenby undirected digraphs G :(1) loop conditions G , where G is bipartite,(2) loop conditions G , where G is non-bipartite and loopless,(3) loop conditions G , where G contains a loop (trivial).Conditions (1) imply (2) imply (3). Conditions (2) are the weakest non-trivialloop conditions.Proof. If a graph G is bipartite (contains a loop, respectively), then the G loopcondition is equivalent to the edge loop condition by item 1 of Proposition 2(is trivial, respectively). If G is non-bipartite and loopless, then the G loopcondition implies a clique loop condition (item 5 of Proposition 2), which impliesthe triangle loop condition (by Proposition 4), which implies the l -cycle loop7ondition for any odd l (by Proposition 3), which, finally, implies the G loopcondition (by item 4 of Proposition 2).Clearly, conditions (1) imply (2) imply (3), and (3) do not imply (2). Theimplication from (2) to (1) cannot be reversed even for idempotent finitelygenerated varieties: an example of an algebra satisfying (2) but not (1) is thealgebra ( { , , } , m ) where m ( x, y, z ) = x + y − z modulo 3.Theorem 1 also provides an alternative proof to the fact [10] that the exis-tence of a near unanimity term implies the existence of a Siggers term. Recallthat a near unanimity (NU) term is an n -ary term t satisfying the identity t ( x, . . . , x, y, x, . . . , x ) = x for all positions of y . Theorem 2.
If an algebra (or a variety) has an n -ary NU term, then it alsohas the Siggers term.Proof. Let t be the NU term. The value t ( x , x , . . . , x n ) may be expressed inthe following two ways. t ( t ( x , x , x , . . . , x ) , t ( x , x , x , . . . , x ) , ) , . . . , t ( x n , . . . , x n , x n , x ))= t ( t ( x , x , x , . . . , x n ) , t ( x , x , x , . . . , x n ) , ) , . . . , t ( x , . . . , x n − , x n − , x n )) . This identity may be interpreted as a loop identity (whose graph is the n -clique)for a composed n -ary term. Therefore the n -clique loop condition is satisfiedand then also the triangle loop condition is satisfied since all the loop conditionsgiven by non-bipartite loopless graphs are equivalent. A digraph G is said to be smooth if every vertex has an incoming and an outgoingedge; G is said to have algebraic length 1 if there is no graph homomorphismfrom G to a directed cycle of length greater than one. The following theoremholds for finite algebras [7]. Theorem 3.
Let A be a finite algebra. Let G , H be weakly connected smoothdigraphs of algebraic length one. Then A satisfies loop condition G if and onlyif it satisfies loop condition H . Less formally, all connected smooth loop conditions with algebraic length 1are equivalent for finite algebras. This gives us a simpler weakest loop conditionthan the triangle for finite algebras: s ( a, r, e, a ) = s ( r, a, r, e ).For general varieties it is no longer true that all such loop conditions areequivalent. However, very recently, we have shown that all loop conditionsgiven by strongly connected graphs of algebraic length 1 are. These results willappear in a forthcoming paper. References [1] Libor Barto and Marcin Kozik. Absorbing subalgebras, cyclic terms, andthe constraint satisfaction problem.
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