Equationally complete classes of commutative monoids with monus
Let \((M,+,0)\) be a commutative monoid such that the relation \(a\leq b\) iff \(b=a+c\) for some \(c\in M\) is a partial order on M. Suppose that for any a,\(b\in M\) there exists a smallest \(x\in M\), \(x=a\dot-b\) with \(a\leq b+x\); then M is called a commutative monoid with monus (CMM). \textit{B. Bosbach} [Math. Ann. 161, 279-295 (1965; Zbl 0154.017)] has proved that the class of all CMM's forms a variety. The results of this paper are concerned with this variety and are the following: 1) The variety \(V({\mathbb{N}})\) of CMM's generated by the natural numbers \(({\mathbb{N}},+,0)\) with \(a\dot-b=a-b\) if \(a\geq b\) and \(=0\) otherwise consists of the positive cones of arbitrary lattice ordered groups. 2) The variety \(V(2)\) of CMM's generated by the two element Boolean algebra \((B,\wedge,\vee)\) with \(a+b=a\vee b\) and \(a\dot-b=a\wedge b'\) consists of all Boolean algebras and all prime ideals of arbitrary Boolean algebras. 3) \(V({\mathbb{N}})\) and V(2) are the only equational complete varieties of CMM's.
scientific article; zbMATH DE number 3875544