On pushing out frames
Bernhard Banaschewski · Czech digital mathematics library · 1990
This paper deals with the preservation of monomorphisms by pushouts in the category of frames.It shows that, for a frame L, pushout along every u: L -• M preserves monomorphisms iff the congruence frame of L is Boolean, and derives several further consequences.Among the lemmas needed, it is proved that, for regular L, the coequalizer of any f t g:L -• M is the map M -• |s, x ~~> sevs, for s = V/(x)^g(y) («-y = 0) Keywords: Frame, pushout, preservation of monomorphisms by pushouts, Boolean con gruence frame Classification: 54D30, 54H99 Among the question discussed at one of the problem sessions during the Con ference on Locales and Topological Groups in Curasao in August 1989 was the following:For pushouts V L > N (*) J. _ 1-M --> P in the category of frames, what conditions will ensure that v is monic whenever v is monic?As it stands, this question permits various specific interpretations, depending on the nature of the conditions envisaged.Thus, one might have in mind the possibility of conditions involving both, u and v, as in the fairly obvious observation, based on Stone Duality for finite distributive lattices, that the desired conclusion holds whenever L, A.f, and N are finite.On the other hand, one may consider the case that focusses on L and ask:For which frames L does pushout along every homomorphism u: L -• M pre serve monomorphisms?This is the question which is settled in this note.We recall a number of basic notions.A frame is a complete lattice L satisfying the distribution law .V5 = Va~*(*Є 5) (a€L, 5 C L),