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.
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Martinez, J., Zenk, E.R. When an algebraic frame is regular. Algebra univers. 50, 231–257 (2003). https://doi.org/10.1007/s00012-003-1841-1
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DOI: https://doi.org/10.1007/s00012-003-1841-1