Cellular lattices (Q1300320)

It is well known that every algebraic lattice is upper-continuous (i.e. \(a\wedge\bigvee A=\bigvee(a\wedge b:b\in A)\) for all upper-directed subsets \(A)\). The converse implication is in general not true. The author shows that in some well-known theorems [see \textit{P. Crawley} and \textit{R. P. Dilworth}, Algebraic theory of lattices (Prentice-Hall, Englewood Cliffs, N.J.) (1973; Zbl 0494.06001)] the hypothesis ``algebraic lattice'' can be replaced by ``upper-continuous lattice''. Assume that \(K\) denotes a fixed class of lattices. A lattice \(L\) is said to be upper \(K\)-cellular, whenever for any two different covers \(b,c\) of an element \(a\), the interval \([a,b\vee c]\) is isomorphic to a member of \(K\). Lower \(K\)-cellular is defined dually. \(K\)-cellular means both lower and upper \(K\)-cellular. Finally, \(M_n\) denotes the \((n+2)\)-element modular lattice with \(n\) atoms. The main results: (1) Let \(L\) be a strongly atomic upper-continuous lattice. Then (a) \(L\) is modular iff it is weakly upper semimodular and weakly lower semimodular; (b) \(L\) is modular iff \(L\) is \(\{M_n:n\) is a cardinal number\}-cellular; (c) \(L\) is distributive iff \(L\) is \(\{M_2\}\)-cellular; (d) \(L\) is locally distributive (i.e. \([a,u_a]\) is distributive for every \(a\in L\) and \(u_a\) is join of elements covering \(a)\) iff \(L\) is upper \(\{M_2\}\)-cellular. (2) The lattice \(\text{Eq}(X)\) of all equivalence relations on a set \(X\) is upper \(\{M_2,M_3\}\)-cellular and lower \(\{M_2,M_3,\text{Eq}(\{1,2,3,4\})\}\)-cellular.