Varieties of lattice-ordered algebras
The variety of lattice-ordered groups generated by fully ordered groups is axiomatised either by (i) \((x\vee y)^ 2=x^ 2\vee y^ 2\) or (ii) \((x\vee 1)\wedge y^{-1}(x^{-1}\vee 1)y=1\). The variety of lattice- ordered loops generated by fully ordered loops is axiomatised by: (a) \((x/z\vee 1)\wedge y\backslash((z/x\vee 1)y)=1\) (b) \((x/z\vee 1)\wedge(((z/x\vee 1)y)t)/yt=1\) (c) \((x/z\vee 1)\wedge(ty\backslash(t(y(z/x\vee 1))))=1.\) This shows that the associative law can largely be dispensed with - an interesting result. The other important theorem in this paper is that every lattice-ordered loop is distributive. The authors provide a start to the study of lattice-ordered loops and challenge others to continue it.
scientific article; zbMATH DE number 3849297
scientific article