A special class of congruences on ${\kappa}$-frames

Abstract

J. Madden has shown that in contrast to the situation with frames, the smallest dense quotient of a ${\kappa}$-frame need not be Boolean. We characterise these so-called ${d}$-reduced ${\kappa}$-frames as those which may be embedded as a generating sub- ${\kappa}$-frame of a Boolean frame. We introduce the notion of the closure of a ${\kappa}$-frame congruence and call a congruence clear if it is the largest congruence with a given closure. These ideas are used to prove ${\kappa}$-frame analogues of known results concerning Boolean frame quotients. In particular, we show that d-reduced ${\kappa}$-frames are precisely the quotients of ${\kappa}$-frames by clear congruences and that every ${\kappa}$-frame congruence is the meet of clear congruences.