Epic substructures and primitive positive functions

CAMPERCHOLI, MIGUEL

Buenos Aires

Congreso; XXI Coloquio Latinoamericano de Álgebra; 2016

UBA

The structure A is an epic substructure of B in the class K provided that forevery pair of homomorphisms h,h´:B -> C, with C in K, and such that h and h´agree on A, we have that h = h´. For example, either of the 3-element sublattices of 2 × 2 is an epic substructure of 2 × 2 in the class of distributive lattices. Thefollowing result characterizes epic substructures in algebraic terms. Recall that a primitive positive (p.p.) formula is an existencial formula whose matrix is a conjunction of atomic formulas.Theorem 1. Let K be a class closed under ultraproducts and A ≤ B structures.T.f.a.e.:- A is an epic substructure of B in K.- For every b in B there are a p.p. formula F(x,y) and v in A^n such that: B satisfies F(v,b) F defines a (partial) function in K.The above theorem says that A is an epic substructure of B in K in B is "gener-ated" by A through primitive positive definable partial functions. Thus it provides an algebraic explanation to the epicness phenomenon. In our talk we shall discuss some applications of Theorem 1 as well.