Characterizations of maximal-sized n-generated algebras

Transcription

1 Characterizations of maximal-sized n-generated algebras Joel Berman Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago AMS Special Session on Universal Algebra and Lattice Theory University of Hawaii at Manoa March 3 4, 2012

2 Definition: For n a positive integer and K a finite set of finite algebras of the same similarity type, let L(n, K) denote the largest n-generated subdirect product whose subdirect factors are algebras in K. For a finite algebra G generated by X = {x 1,..., x n } with G subdirectly embedded in A U, code each coordinate u U as a function u : X A where u(x i ) = x i u for x i the image of x i. u(x ) generates all of A. u : X A such that u(x ) generates A is called a valuation. val(x, A) denotes the set of all valuations of X to A. For u val(x, A), let Alg(u) denote A, the target algebra of u G is subdirectly embedded in A val(x,a).

3 Primal Cluster Theorem. If K = {A 1,..., A k } is a finite set of finite, nontrivial, pairwise nonisomorphic algebras of the same similarity type with V = HSP K, then for all n 0: L(n, K) F V (n) k A i A i n [Birkhoff]. Moreover, the following are equivalent. 1. (bound obtained): L(n, K) = k i=1 A i A i n for all n (algebraic): Each A i is simple, rigid, has no proper subalgebras, and V is arithmetical. [Pixley] 3. (computational): L(m, K) = k i=1 A i A i m for m = max{2, k}. [Sioson] [Sierpinski] i=1

4 Theorem: Suppose V is locally finite, X = {x 1,..., x n }, and S n = {S 1,..., S k } is a set of n-generated, pairwise nonisomorphic, algebras in V. Let U be a slim set of valuations for the set of all valuations from X to the algebras in S n. Let U i denote all u U with Alg(u) = S i. If each Con S i is linearly ordered, then L(n, S n ) k ω(s i, Con S i, U i ). i=1 Moreover, if S n is the set of all n-generated, subdirectly irreducible algebras in V and n 3, then TFAE:

5 1. (bound obtained): F V (n) = L(n, S n ) = k i=1 ω(s i, Con S i, U i ). 2. (algebraic): V is arithmetical and for all v w U { Sg Alg(v) Alg(w) Cg Alg(w) (vw) if Alg(v) = Alg(w); (vw) = Alg(v) Alg(w) if Alg(v) Alg(w). 3. (computational): Let W be a slim set of valuations to S 3 with X = 3. For S S 3, W S := {w W : Alg(w) = S}. L(3, S 3 ) = Sg Alg(v) Alg(w) (vw) = S S 3 ω(s, Con S, W S ) { Cg Alg(w) (vw) Alg(v) Alg(w) and for all v w U if Alg(v) = Alg(w); if Alg(v) Alg(w)

6 Suppose A is a finite algebra, U val(x, A) and C = {1 A = θ 0 > θ 1 > > θ m > θ m+1 = 0 A.} Con A. Code the congruence classes of θ l as strings of l integers i 1... i l. r(ɛ) denotes the number of θ 1 classes in A, the unique θ 0 class. r(i 1... i l 1 ) denotes the number of θ l classes in the θ l 1 class with label i 1... i l 1. For each congruence relation θ l define an equivalence relation l on the set U by v l w iff x X, (v(x), w(x)) θ l. Thus, 1 U = 0 1 > m m+1 = 0 U. Code the l equivalence classes as strings of integers j 1... j l. s(ɛ) denotes the number of 1 classes in U, the unique 0 class. s(j 1... j l 1 ) denotes the number of l classes in the l 1 class with label j 1... j l 1. With this notation ω(a, C, U) = s(ɛ) r(ɛ) j 1 =1 i 1 =1 s(j 1...j l 1 ) j l =1 r(i 1...i l 1 ) i l =1 s(j 1...j m 1 ) j m=1 r(i 1...i m 1 ) i m=1 r(i 1... i m ) s(j 1...j m).

7 Examples ω(a, C, U) = s(ɛ) r(ɛ) j 1 =1 i 1 =1 s(j 1...j l 1 ) j l =1 r(i 1...i l 1 ) i l =1 s(j 1...j m 1 ) j m=1 r(i 1...i m 1 ) i m=1 r(i 1... i m ) s(j 1...j m). If all θ l C are uniform, then all r(i 1... r l ) = r l. If all l are uniform, then all s(j 1... j l ) = s l. Then ω(a, C, U) = ( r0 ( r1... (r m 2 (r m 1 r sm m ) s m 1 ) s m 2... ) s 1 ) s0. Suppose Con A is a chain and all congruences are uniform and that val(x, A) = A X. Then the l are uniform equivalence relations. In fact, s l = rl n for r l and s l as above. Then ω(a, C, A X ) = m l=0 r r n 0 r n 1 r n l l.

8 ω(a, C, U) = s(ɛ) r(ɛ) j 1 =1 i 1 =1 s(j 1...j l 1 ) j l =1 r(i 1...i l 1 ) i l =1 s(j 1...j m 1 ) j m=1 r(i 1...i m 1 ) i m=1 r(i 1... i m ) s(j 1...j m). If A is a simple algebra, then it can be argued that a slim set of coordinates for the set val(x, A) has cardinality val(x,a) Aut A. If A is simple, then m = 0 in the formula for ω(a, Con A, U). So ω(a, Con A, U) = A U. If V is an n-semisimple variety, the theorem gives TFAE: (bound obtained): F V (n) = L(n, S n ) = S S n S (algebraic): V is arithmetical and for all v w U Sg Alg(v) Alg(w) (vw) = Alg(u) Alg(w). (computational): F V (n) = val(x,s) S S 3 S Aut S and for all val(x,s) Aut S. v w U, Sg Alg(v) Alg(w) (vw) = Alg(u) Alg(w).

9 Talking points: Use the computational characterization to determine whether or not the bound obtained characterization holds for n n 0. With the UACalc software, use ω(a, C, U) to halt a computation when the upper bound is obtained. Thereby avoid the time consuming check that no new elements will be generated. Use ω(a, C, U) and the l equivalence relations to decompose a large computation into overlapping manageable chunks. For a given chain C of equivalence relations on A construct a congruence primal algebra with Con A = C and with ω(a, C, A X ) n-ary term operations. What about rectangular subuniverses? What about congruence relative Stone varieties? What about the complexity of deciding if HSPA is arithmetical?