Convexities of normal valued lattice ordered groups
Ján Jakubík · Czechoslovak Mathematical Journal · 2000
Convexities of lattice ordered groups were investigated in [5].Earlier, convexities of lattices and of d-groups had been dealt with in [3] or [4], respectively.Let us recall that the notation of convexity of lattices was introduced by Fried ([7], p. 225).We denote by G -the class of all lattice ordered groups; A -the class of all abelian lattice ordered groups; N -the class of all normal valued lattice ordered groups; X 0 -the class of all one-element lattice ordered groups.For G ∈ G we denote by C(G) the convexity of lattice ordered groups which is generated by G. Let Z, Q and R be the additive group of all integers, rationals and reals, respectively, with the natural linear order.If we consider a result on varieties of lattice ordered groups, torsion classes or radical classes, then we can ask whether a similar result holds for convexities.The following result is well-known (cf., e.g., [1]):(A) There exists a variety X 1 (namely,A result analogous to (A) holds neither for torsion classes nor for radical classes.In the present paper we prove:(B) There exists a convexity Z 1 = X 0 (namely, Z 1 = C(R)) such that, whenever Z is a convexity with X 0 = Z ⊆ N , then Z 1 ⊆ Z.Some further results are also proved.Let us remark that the class N is large in the sense that whenever V is a variety with V = G , then V ⊆ N (cf., e.g., [1]).