The lattice of $R$-subalgebras of a bounded distributive lattice
L. Vrancken-Mawet · Czech digital mathematics library · 1984
Using Priestley's duality, we investigate tbm lattice S R (L) of the -[0,1$-8UDlattlees of a given bounded distributive lattice L which are closed under relative compplementation.We characterize those bounded distributive lattices L such that S R (L) is semimodular, modular, distributive or Boolean., Key words; Distributive lattice -Relative complementati on -Priestley's duality -Congruences on partially ordered spaces.Classification: 06DO5.Introduction.In his study on Boolean lattices R-genera~ ted by distributive lattices, Gratter considers particular ^O^J-sublattices of a given bounded distributive lattice, namely those which are closed under relative complementation* The purpose of this paper is to study these sublattices, which we call R-subalgebras.It turns out that Priestley's duality is a well-adapted tool to achieve this alnu In Section 1, we introduce the concept of congruence on a Priestley space, which is dual to that of R-subalgebraj the lattice of all R-subalgebras of a bounded distributive lattice is dually isomorphic to the lattice Con(X) of all congruences on the dual X of L. Section 2 is devoted to th.e &tшţy of GJCЛCX).ІЗ* ÇÄГІlcular we cћaracteгize those Priestley spaces whoэe congruence lattice is semi-modular, mođular or diзtributive respectively.We trenslate these results ln terms of R-aubölgebras in Section 3* We adopt standard set theoretic notations.Let us however recall some of them.For set X, we denote by |XІ its cardinal andЪy Eq(X) its equivalence lattice.If ö€Eq(x), XGX nd Ecx, we write x G for the -class of x nd E » Ufx ö I x €E$; E is -вatttтвtвÖ if E -= E. If X =-(X,ᣠ) is a poset, pЧq means that q coverв p and p H q means that p nd q re not compaг ble.We say that Ecx is convex if x-éztćy nd x,ycE imply that sьбE.An oгder connected component (o.c.c.) of X is a subset E of X which is minimal with respect to the property of Ъeing Ъoth ţfc* creasing and decre sing.Finally, the n-element chain is denoted by n.