Saturated algebras in filtral varieties (Q1088704)

An algebra A in a class \({\mathcal K}\) is said to be saturated if every injective epimorphism \(A\to B\) (with \(B\in {\mathcal K})\) is an isomorphism. A variety \({\mathcal V}\) is balanced if every member is saturated. In this paper, it is proved that if \({\mathcal V}\) is a filtral variety, then \[ \Gamma ^ a({\mathcal M}^ +_{sat})\subseteq {\mathcal V}_{sat}\subseteq \Gamma ^ a({\mathcal M}^ +). \] Here, \({\mathcal M}^ +\) is the class of simple and trivial algebras of \({\mathcal V}\), \({\mathcal K}_{sat}\) is the class of saturated members of \({\mathcal K}\) and \(\Gamma ^ a({\mathcal K})\) denotes the class of boolean products of members of \({\mathcal K}\). As a corollary, we have that a balanced filtral variety is a discriminator variety, and that a discriminator variety is balanced if and only if every simple algebra is saturated in \({\mathcal M}^ +\). Examples are constructed that demonstrate that the inclusions in the main theorem may be strict.