The ideal structure of simple ternary algebras

On $${\chi _{_{mub} } }$$ -latticesand convex substructures of lattices and semilattices

“…A further study about extensions and generalizations will be the topic of a subsequent paper. We close this section by The elements of Csub (S) are convex subtrees of S. In Theorem 4 of [7] it is shown that these constitute a median semilattice, which is a distributive Z~b-lattice as stated above.…”

The subtree center of a tree

“…Consider a star of five points v 1 , v 2 , v 3 , v 4 , v 5 with v 4 as a middle point, join to this star a path v 5 v 6 v 7 v 8 , and denote the tree thus obtained by T. The tree T shows that one of the two points of a centroid need not be a least L-eccentricity point of a tree: In T, the centroid is {v 4 …”

“…A tree may contain several central subtrees. This can be seen by considering a path of six points: e L (͗{v 3 4 shows that there may be several least central subtrees in a tree: e L (͗{v 3 …”

“…The subtrees (convex subalgebras/ideals) of a tree constitute a median algebra (see [1,Thm. 2.7] or [4,Thm. 4]) and, thus, the joinsemilattice L(T) has a limited distributivity property (a median algebra property).…”

Least central subtrees, center, and centroid of a tree

“…The joinsemilattice of subtrees was introduced in [11]; see also [10]. The underlying graph G L is a median graph [1,7,9].…”