This is a summary of some of the main results in the monograph Faithfully Ordered Rings (Mem. Amer. Math. Soc. 2015), presented by the first author at the ALANT conference, Będlewo, Poland, June 8-13, 2014. The notions involved and the results are stated in detail, the techniques employed briefly outlined, but proofs are omitted. We focus on those aspects of the cited monograph concerning (diagonal) quadratic forms over preordered rings.
In an -group with an appropriate operator set it is shown that the -value set can be embedded in the value set . This embedding is an isomorphism if and only if each convex -subgroup is an -subgroup. If has a.c.c. and is either representable or finitely valued, then the two value sets are identical. More generally, these results hold for two related operator sets and and the corresponding -value sets and . If is a unital -ring, then each unital -module over is an -module...
This paper deals with some properties of n-fold commutative ideals and n-fold weak commutative ideals in BCK-algebras. Afterwards, we construct some algorithms for studying foldness theory of commutative ideals in BCK-algebras.
“The kernel functor” from the category of archimedean lattice-ordered groups with distinguished weak unit onto LFrm, of Lindelöf completely regular frames, preserves and reflects monics. In , monics are one-to-one, but not necessarily so in LFrm. An embedding for which is one-to-one is termed kernel-injective, or KI; these are the topic of this paper. The situation is contrasted with kernel-surjective and -preserving (KS and KP). The -objects every embedding of which is KI are characterized;...
In this paper we have given the construction of free -groups generated by a po-group and the construction of free products in any sub-product class of -groups. We have proved that the -free products satisfy the weak subalgebra property.