Algebraic extensions of an archimedean lattice-ordered group, I

Abstract

Within a quite general class of structures, it is shown (pursuing a lead of Bacsich) that an extension A≤ B is algebraic according to a definition transplanted from a model-theoretic context of Jónsson (we say, ‘an AE’) if and only if A≤ E≤ B implies A≤ E is epimorphic in the categorical sense. This is valid in the category Arch of archimedean l-groups, with l-homomorphisms, and its subcategory W , whose objects have distinguished weak unit, whose morphisms preserve the weak unit. Here, we focus on W (turning to Arch in a sequel; an understanding of W is necessary first), deploying our theory of epimorphisms developed previously. We find the forbidding pieces of pathology than an AE of an AE need not be an AE, and only rarely does an object have an AE which is also algebraically closed (AC). But still, there are canonical extensions A≤c 3A≤C[ C A]≤C[ C Aδ] with the features: A≤ B is an AE if and only if the ideal generated by A in B embeds into c 3 A; every AE of A embeds into C[ C A ] and C[ C A ] is minimum for that; C[ C Aδ ] is AC, and minimum for that plus containing every AE of A—so A is AC if and only if A = C[ C Aδ] . The defect that AE's are not composable can be attributed to the fact that there are AE's A≤ B with elements of B not order-dominated by A; defining these away produces a ‘perfect theory’. Corollaries of the development are that the c 3 objects constitute the least reflective subcategory with each reflection an essential extension, and that this class coincides with the class of extremal subobjects of AC objects.