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.
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Presented by W. McGovern.
Financial assistance from the National Research Foundation of South Africa is acknowledged.
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Manuell, G. A special class of congruences on \({\kappa}\)-frames. Algebra Univers. 78, 125–130 (2017). https://doi.org/10.1007/s00012-017-0439-y
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DOI: https://doi.org/10.1007/s00012-017-0439-y