Epic subalgebras via primitive positive functions

CAMPERCHOLI, MIGUEL

Buenos Aires

Congreso; 3rd MaToMUVI Meeting, Buenos Aires, febrero de 2015.; 2015

Let A and B be algebras and K a class of algebras. The algebra A is an epic subalgebra of B in K provided that for every pair of homomorphisms h,g from B to C, with C in K, and such that h and g agree on A, we have that h = g. For example, either of the 3-element sublattices of 2×2 is an epic subalgebra of 2×2 in the class of distributive lattices. The following result characterizes epic subalgebras in algebraic terms. Recall that a primitive positive (pp) formula is an existencial formula whose matrix is a conjunction of atomic formulas. Theorem: Let K be a first order axiomatizable class and A a subalgebra of B. T.f.a.e.: (1) A is an epic subalgebra of B in K. (2) For every b in B there are a pp formula F(x,y) and a from A such that: (i) B |= F(a,b) (ii) F defines a function in K. The above theorem says that A is an epic subalgebra of B in K iff B is "generated" by A through primitive positive definable partial functions. Thus it provides an algebraic explanation to the epicness phenomenon. Epic subalgebras in filtral varieties are investigated in [1]. As an application of Theorem 1 we generalize several results in that paper to filtral quasivarieties, providing new straightforward proofs. References [1] Clifford Bergman, Saturated algebras in filtral varieties, Algebra Universalis 24 (1987), Issue 1-2, pp. 101--110.