Irredundance

Abstract

Clearly Irr A ≤ |A|. If A is a subalgebra of B, then Irr A ≤ Irr B, and Irr can change to any extent from B to A (along with cardinality). The same is true for A a homomorphic image of B. Concerning the derived operations, we note just the obvious facts that Irr _S+ A = IrrA, Irr _S- A = ω, Irr_h-A = ω, and _d Irr_s +A = IrrA. Obviously any chain is irredundant; so Length A ≤ IrrA. The difference can be large, e.g. in a free BA. By Theorem 4.25 of Part I of the BA handbook, πA ≤ IrrA. In particular, if |A| is strong limit, then |A| = IrrA, since then πA = |A|.