Residuation in commutative ordered monoids with minimal zero (Q2752408)

The authors define a new algebraic system called semi-integral residuated commutative monoid (sircomonoid). An algebra \(B= \langle B;\oplus,\mathop{\dot-},0\rangle\) of type \(\langle 2,2,0\rangle\) is called a sircomonoid if it satisfies that for \(a,b,c\in B\), (i) \(\langle B;\oplus,0\rangle\) is a commutative monoid; (ii) the relation \(\leq\) defined by \(a\leq b\) iff \(a\mathop{\dot-} b=0\) is a partial order of \(B\); (iii) \(a\mathop{\dot-} b\leq c\) iff \(a\leq c\oplus b\); (iv) \(a\leq 0\) implies that \(a= 0\).NEWLINENEWLINENEWLINEThey consider fundamental properties of these algebras:NEWLINENEWLINENEWLINE1. Every BCI-algebra is a subalgebra of the reduct \(\langle B;\mathop{\dot-},0\rangle\) of a sircomonoid \(B= \langle B;\oplus,\mathop{\dot-},0\rangle\). Thus BCI-algebras are exactly the residuation subreducts of sircomonoids;NEWLINENEWLINENEWLINE2. The class \({\mathbf S}{\mathbf I}{\mathbf R}{\mathbf C}{\mathbf O}{\mathbf M}\) of all sircomonoids and the class \({\mathbf B}{\mathbf C}{\mathbf I}{\mathbf A}\) of all BCI-algebras are relatively congruence modular systems;NEWLINENEWLINENEWLINE3. \({\mathbf S}{\mathbf I}{\mathbf R}{\mathbf C}{\mathbf O}{\mathbf M}\) has \(2^{\aleph_0}\) subvarieties;NEWLINENEWLINENEWLINE4. \({\mathbf B}{\mathbf C}{\mathbf I}{\mathbf P}\) has a local deduction detachment theorem, where \({\mathbf B}{\mathbf C}{\mathbf I}{\mathbf P}\) means an assertional logic of \({\mathbf B}{\mathbf C}{\mathbf I}{\mathbf A}\), that is, a deductive system which has \({\mathbf B}{\mathbf C}{\mathbf I}{\mathbf A}\) as an equivalent quasivariety semantics.NEWLINENEWLINENEWLINEThis paper is very interesting and valuable to the progress of algebraic logic.