A classification of hull operators in archimedean lattice-ordered groups with unit

Abstract. The category, or class of algebras, in the title is denoted by
W. A hull operator (ho) in W is a reflection in the category consisting of
W objects with only essential embeddings as morphisms. The proper class
of all of these is hoW. The bounded monocoreflection in W is denoted B.
We classify the ho’s by their interaction with B as follows. A “word” is a
function w : hoW −→ WW obtained as a finite composition of B and x a
variable ranging in hoW. The set of these,“Word”, is in a natural way a
partially ordered semigroup of size 6, order isomorphic to F(2), the free 0 −1
distributive lattice on 2 generators. Then, hoW is partitioned into 6 disjoint
pieces, by equations and inequations in words, and each piece is represented
by a characteristic order-preserving quotient of Word (≈ F(2)). Of the 6: 1
is of size ≥ 2, 1 is at least infinite, 2 are each proper classes, and of these 4,
all quotients are chains; another 1 is a proper class with unknown quotients;
the remaining 1 is not known to be nonempty and its quotients would not be
chains.

A classification of hull operators in archimedean lattice-ordered groups with unit Keywords:

lattice-ordered group , Archimedean , weak unit , bounded monocoreflection , essential extension , hull operator , partially ordered semigroup