When an algebraic frame is regular

Abstract

It is shown that an algebraic frame L is regular if and only if its compact elements are complemented. More generally, it is shown that each pseudocomplemented element is regular if and only if each \( c^{\bot\bot} \), with c compact, is complemented. With a mild assumption on L, each \( c^{\bot} \), with c compact, is regular precisely when \( p \bigvee q = 1 \) for any two minimal primes p and q of L. These results are then interpreted in various frames of subobjects of lattice-ordered groups and f-rings.