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This paper explores the equational complexity of graph algebras and investigates whether the triangle algebra is linearly or quadratically bounded. The research was presented at the Conference on Universal Algebra and Lattice Theory in Szeged, Hungary in 2012, celebrating the achievements of Béla Csákány.
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Equational Complexity of Graph Algebras(Is the Triangle Algebra Linearly or Quadratically Bounded?) At the Conference on Universal Algebra and Lattice Theory in Szeged, Hungary June 21-24, 2012 celebrating the achievements of Béla Csákány at his 80th birthday Zoltan Szekely Division of Mathematical Sciences University of Guam, Mangilao, GU 96923, USA zoltan_szekely@yahoo.com
Characterization of finitely based graph algebras Proved by using a technique called the shift automorphism method.
Inherently nonfinitely based algebras • A variety V is called inherently nonfinitely based if • V is locally finite and • V is included in no finitely based, locally finite variety.
Equational complexity results • Sz., 1998: finite algebras with equational bounds dominating strictly monotone increasing ‘sublinear’ functions; including versions of graph algebras and automatic algebras. A finite version of the shift automorphism method was employed on a ‘Moebius Torus’ structure.
Equational complexity results • George McNulty, 2004: the same algebras have equational bounds dominating some strictly increasing linear functions. • Sz., 1998: the equational bounds of finite graph algebras are dominated by quadratic functions; equational bounds of Lyndon’s algebra and some other finite automatic algebras are dominated by linear functions. • Gabor Kun, Vera Vertesi, 2004: exhibit finite algebras for any d such that the equational bounds on these algebras are about a polynomial in degreed; • Marcin Kozik, 2004: finds finite algebra with 2-EXP equational complexity; • Marcel Jackson, George McNulty, 2010: the equational complexity of Lyndon’s automatic algebra lies between n-4 and 2n+1.
Equational bound: constant bound on a nonfinitely based finite algebra • If a variety is finitely based then its equational bound is dominated by a constant. • Limitation on the shift automorphism method: • Ross Willard, 2005: V a variety generated by a finite algebra of finite type, if V is inherently nonfinitely based by reason of the shift automorphism method, then the equational bound of V cannot be dominated by a constant function. • Applies to most of the known examples of inherently nonfinitely based finite algebras.
The variety generated by an inherently nonfinitely based finite algebra … … A
Birkhoff basis and Birkhoff bound • Let V be a locally finite, nonfinitely based variety. • The n-generated free algebra of V is finite, so the n-variable equational theory of V is finitely based. Call this finite basis the Birkhoff basis. • In case of groupoids (graph algebras, automated algebras) the Birkhoff basis is the collection of products uv = w where u, v and w are shortest representative terms of the congruence classes of the n-variable term-algebra of V . • Take an n-element input algebra Bto the finite algebra membership problem of V . To decide the membership of B we only need to check equations with no more than n variables. Use the Birkhoff basis. • The length of equations in the Birkhoff basis depends on the length of the shortest representative terms. We obtain a corresponding equational bound. • The equational bound obtained this way is called a Birkhoff bound on V .
Graph algebras: length of representative term is proportional to the number of edges in the term graph • The Birkhoff bound depends on the number of edges in the different versions of term graphs corresponding to the shortest representative terms in the congruence class of the n-variable term algebra of the given variety. • The largest of these edge numbers for a given n provides estimation to both the computational and the equational complexities of the corresponding finite algebra membership problem with an input algebra of size n.
Equational bounds on nonfinitely based graph algebras • Let G be a finite graph • Loops are allowed, multiple edges are not • The corresponding graph algebra is • If G has no induced subgraph isomorphic to any of the four forbidden graphs, then is finitely based and therefore its equational bound is dominated by a constant function. • If G has an induced copy of any of the four forbidden graphs, then the variety V=HSP( ) contains a subvariety W generated by one of the following four algebras: • As a consequence, the Birkhoff bound on V is at least as much as the Birkhoff bound on W. • Therefore we only need to check the Birkhoff bounds on the above mentioned four varieties.
0. The equational theory of • Theorem: • Term graphs may have up to edges. • Corollary: the Birkhoff bound on graph algebras is no more than • In particular, the graph algebra of any finite graph is 3n(n-1)+4 bounded.
1. The equational theory of • Theorem (Shallon, 1979): • The number of edges of the largest n-variable loopless term graph corresponding to a representative term is proportional to . • Corollary: the Birkhoff bound on the graph algebra is .
2. The equational theory of • Theorem (Murskii, 1965) • The number of edges of the largest n-variable reduced term graph corresponding to a representative term is proportional to . • Corollary: the Birkhoff bound on the graph algebra • is .
3. The equational theory of • Zero terms provide the zero value identically by any evaluation in the underlying algebra. • Lemma: A term p is a zero term for if and only if the term graph of p is not 2-colorable. In particular if the term graph has any looped vertex or has a complete (loopless) subgraph on at least 3 vertices, then p is a zero term. • Theorem: An equation belongs to the equational theory of if and only if • both p and q are zero terms; or • the equation is regular and • the graph parts of the term graphs of p and q are the same 2-colorable graph and • the roots of p and q are in the same color class.
3. The equational theory of • Idea of the proof: • Let the elements of • be denoted as follows: • 1. a nonzero evaluation of a term p in may use the elements a, a’, b, b’ to substitute its variables. Leaving out the primes we obtain a 2-coloring of the term graph by the “colors” a and b in the natural way. So must be 2-colorable. • Assume a 2-coloring c of using a and b as colors is also a 2-coloring of , but has an edge e=(x,y) which is not an edge in . W.l.o.g. we may assume that the following substitution was applied to the endpoints of e: ax, by. Now, change the substitution into a’x, b’y. • The resulting mapping c’ still defines a 2-coloring of where {a, a’} and {b, b’} are collapsed into one color class. But it is not a 2-coloring of • . Therefore, in the corresponding evaluation, still evaluates to a non-zero element while evaluates to zero. So fails in .
3. The equational theory of 2-coloring c: using a and b with color classes {a}, {b} 2-coloring c’: using a, a’, b, b’ with color classes {a,a’},{b,b’} • Idea of the proof: a, a’ a, a’ b b b b e a a a a b, b’ b, b’ e does not violate c, but violates c’! neither c nor c’ is violated!
3. The equational theory of • Corollary: An equation holds in if and only if • neither of the term graphs of p and q are 2-colorable graphs; or • both are 2-colorable graphs and • have the same set of vertices and • have their roots in the same color class and • have the same set of edges. • The corresponding multiplication rule for term graphs can be easily established. • The number of edges of the largest n-variable term graph corresponding to a representative term is proportional to . • Theorem: the Birkhoff bound on the graph algebra is .
4. The equational theory of • Lemma: A term p is a zero term for if and only if the term graph of p is not 3-colorable. In particular if the term graph has any looped vertex or has a complete (loopless) subgraph on at least 4 vertices, then p is a zero term. • Theorem: An equation belongs to the equational theory of • if and only if • both p and q are zero terms; or • the equation is regular and • the graph parts of the term graphs of p and q have the same set of 3-colorings and • the roots of p and q are in the same color class by any 3-coloring.
4. The equational theory of • Corollary: An equation holds in if and only if • neither of the term graphs of p and q are 3-colorable graphs; or • both are 3-colorable graphs and • they have the same set of 3-colorings and • have the same set of vertices and • have their roots in the same color class in any 3-coloring. • The corresponding multiplication rule for term graphs can be easily established. • The number of edges of the largest n-variable term • graph of a representative term is less than or equals to . • Theorem: the Birkhoff bound on the graph algebra is • at most .
4. The equational theory of • Can we have a sharper upper bound on the number of edges of term graphs corresponding to the shortest representative terms in the term algebra • of ? • In particular, can we have a linear upper bound? • Example: for a uniquely 3-colorable graph on at least 3 vertices, we only need O(n) many edges: • for any given 3-partition, we can easily build a 2-tree which is • (a) uniquely 3-colorable; • (b) the color classes of its 3-coloring are exactly the blocks of the given 3-partition. • (c) It will have 2n-3 edges.
A uniquely 3-colorable graph (2-tree) on n=9 vertices and m=15 edges the color classes of the corresponding unique 3-coloring a 2-tree, built from triangles
A Galois connection (12/1): relational structure, homomorphism as 3-coloring) • Let G=(V,E) be a graph on n vertices, where n is at least 4, a fixed number, with vertex set V and edge set E. The relational structure A(G)=(V,R) has the same underlying set V and the relation R is given by the set E of edges. • Let T be the triangle graph and A(T) the corresponding relational structure. A mapping P from A(G) to A(T) is a 3-coloring of G if P is a homomorphism. P G T A(T) A(G)
A Galois connection (12/2): relate two Boolean lattices • Let K* denote the set of graphs on n vertices. • The power set P(K*) of K* forms a Boolean lattice by the regular set operations, denoted by S(K*). • For a subset K of K*, let CK denote the set of mappings that are homomorphisms from A(G) to A(T) for all G in K (set of common 3-colorings of all graphs in K). • Note: CK is empty if K has any graph that is not 3-colorable. • Let C* denote the set of mappings from V to the set of vertices of T . • The power set P(C*) of C* forms a Boolean lattice by the regular set operations, denoted by S(C*). • For a subset C of C*, let KC denote the set of those graphs G on V such that any mapping P in C is a homomorphism from A(G) to A(T) (any mapping P is a 3-coloring of G). • Note:the graph on V with no edge, denoted by D, is in KC for any C.
A Galois connection:closure operations (12/3) • CK={mappings P : P is a homomorphism from A(G) to A(T) for all G in K}={mappings P : P is a 3-coloring for all G in K} • KC={graphs G : P is a homomorphism from A(G) to A(T) for all P in C} ={graphs G : all P in C are 3-colorings of G} • Galois property: • C is a subset of CK if and only if K is a subset of KC. • Thus, we have a Galois connection between S(K*) and S(C*). • [K]= KCK and [C]= CKC are closure operations. • Note: If G is in [K] and H is a subgraph of G then H is also in [K]. (Subgraphs do not violate 3-colorings). • So [K] has D , the empty graph, as its smallest element.
A Galois connection (12/4):visualized S(C*) S(K*) K* C* [K]= KCK K CK [C]= CKC K C C KC K C
A Galois connection (12/5):MaxC - the largest graph in KC • Theorem: Let C be a set of mappings from V to the set of vertices of T. Then there is a graph MaxC in KC such that • (a) G is in KC if and only if G is a subgraph of MaxC , • (b) for any mapping P outside of [C], P is not a 3-coloring of MaxC. • This graph can be given in the following way: • if C , then MaxC = (V,E) where E is defined by the relation • if C , then MaxC = the complete graph on n vertices. • (Bar denotes the complement of a relation.)
A Galois connection (12/6):MaxC - the largest graph in KC • E is defined by the relation • Notes: • (i) KerP is the equivalence relation associated with the color classes of the 3-coloring P. • (ii) Any subset of , as a relation, describes the edge set of a graph on V that is 3-colorable by the 3-coloring P. • (iii) The intersection of these complements provides the largest graph that is 3-colorable by all 3-colorings in C.
A Galois connection (12/7): describing the closures • Corollary : Let K be a set of graphs on n vertices and let C=CK. Then • [C]=C, in particular, CG=C{G} is a closed set, • [K]= [D, MaxC]. • If K={G} then we’ll use the MaxG = Max(CG)notation. • Note: K*,the subgraph lattice of , is a Boolean lattice itself. [K] is a sublattice of K*.
A Galois connection (12/8):subgraphs of the complete graph C* S (C*) K*=[D, ] MaxG =MaxC K C=CG [G]=KC=[D,Max G] C G C XG K D
A Galois connection (12/9):the join sub-semilattice belonging to G • Let G be a graph on n vertices. Introduce • XG ={H graph with vertex set V: CH=CG} a subset of [G]. • Easy to check the following properties: if H, K are in K* then • C(HK) = C(H) C(K), C(HK) C(H) C(K). • It follows that XG is closed for but not necessarily closed for . • Thus, XG is a join-subsemilattice of [G]. • Corollary: Let G be a graph on n vertices. Then there is a maximal element MaxG in XG . In particular: • (a) G is a subgraph of MaxG, • (b) MaxG has the largest number of edges among the graphs in XG.
A Galois connection (12/10):example on n=9 vertices G on n=9 vertices with m=15 edges MaxG on n=9 vertices with m=26 edges
A Galois connection (12/11): The minimum edge problem • G: a 3-colorable graph on n vertices. • XG ={H graph with vertex set V: CH=CG} • Find a member of XG with fewest number of edges. • Set up a bound on the number of edges in XG as a function of n. • Is this bound quadratic in n? • Is this bound linear in n?
A Galois connection (12/12): How to leave out edges? • Starting from a 3-colorable graph G, some edges might be left out without changing XG. • In the example on n=9 vertices, starting from MaxG, 11 edges may be left out of the 26, to obtain G with 15 edges, without changing the unique 3-coloring. • Observation: An edge e can be left out of the graph G if and only if in any 3-coloring of G -{e} the endpoints of e are in distinct color classes. • The same condition: Let e=(x,y) be an edge of G. • X(G -{e})=XG if and only if • for any mapping P in C(G -{e}), .
Some references • Kirby Baker, George F. McNulty and Heinrich Werner, Shift automorphism methods for inherently nonfinitely based varieties of algebras, Czech Math J. 39 (114), 1989 • Robert Cacioppo, Non-finitely based pseudovarieties and inherently non-finitely based varieties, Semigroup Forum 47 (1993) • I. M. Isaev, Inherentlynonfinitely based varieties of algebras, Sibir Math 30 (1989) • Marcel G. Jackson, George F. McNulty, The equational complexity of Lyndon’s algebra, Algebra Universalis, to appear • Marcin Kozik, On some complexity problems in algebra, Ph.D. thesis, 2004 • Roger C. Lyndon, Identities in finite algebras, Proc. Amer. Math. Soc. 5 (1954) • Ralph McKenzie, The residual bounds of finite algebras, Int. J. Alg. Comput. 6 (1996) • George F. McNulty, Z. Szekely, Ross Willard, The equational complexity of the finite algebra membership problem, submitted to Intern. J. of Alg. and Comput. 18 (2008)
Another view of graph coloring: what is preserved in the adjacency matrix? Adjacency matrix of G 1 1 2 5 Column reduction by P 2 4 3 3 G T Row reduction by P: obtain the adjacency matrix of T 3-coloring P of G by T The important feature of 3-colorability is preserved in the adjacency matrix.
Another view of a colorful apricot field:what is preserved in the matrix? • Take an about 10x5 colorful matrix over the apricot field, owned by Béla and Rozika, arranged on the kitchen table of their home. • Important features of the apricot are all preserved. Where? • In the delicious apricot preserve! • (In Hungarian: kajszi lekvár!) • I am a witness for the high quality preserved as I was lucky to get a sample. Thank you, Béla!
