Doubling convex sets in lattices and a generalized semidistributivity condition
A useful construction for lattices is doubling a convex subset I of a lattice L, i.e., replacing I by \(I\times 2\). In case I is an interval [a,b], this construction was introduced in 1970 by Alan Day to give a simple solution of the word problem in free lattices. Since then, this construction has found various other applications in lattice theory. In the present paper, it is shown that doubling a convex subset preserves the following generalized semidistributivity condition: (C) for all a,b,c\(\in L\), if \(ab=ac\) and \(c+b=c+a\) then \(a\leq c.\) Furthermore, varieties of lattices in which every lattice satisfies (C) are characterized equationally. These results are analogues of a classic series of theorems about doubling lower pseudo-intervals \(I=\cup [a,b_ i]\), i.e., unions of intervals with a common least element, or upper pseudo-intervals, defined dually.
scientific article; zbMATH DE number 4137827
scientific article