Extended-order algebras
A weak extended-order algebra is a triplet \((A,\rightarrow ,1)\) of type \( (2,0)\) such that for all \(a,b,c\in A,\) the following conditions are satisfied: \(a\rightarrow 1=a\rightarrow a=1,\) if \(a\rightarrow b=b\rightarrow a=1\) then \(a=b\), and if \(a\rightarrow b=b\rightarrow c=1\) then \(a\rightarrow c=1.\) If \(a\rightarrow b=1\) then \((c\rightarrow a)\rightarrow (c\rightarrow b)=1\), and if \((b\rightarrow c)\rightarrow (a\rightarrow c)=1\) then \(A\) is called extended-order algebra (as example we have Hilbert algebras and BCK-algebras). For any weak extended-order algebra \(A,\) the relation \(a\leq b\) if and only if \(a\rightarrow b=1\) is an order relation in \(A\) (called natural ordering in \(A).\) Moreover, \(1\) is the top element in \((A,\leq )\). In this paper, several classes of extended-order algebras are considered that lead to most well-known multiplicative ordered structures by means of adjunction, once the completion process due to MacNeille is applied. In particular, complete distributive extended-order algebras are considered as a generalization of complete residuated lattices to provide a structure that suits quite well for many-valued mathematics.
scientific article; zbMATH DE number 5380984
scientific article