Completion and finite embeddability property for residuated ordered algebras.
This is a long, well-written and very interesting paper. A residuated ordered algebra (of type \(T\)) (roa, for short) is a structure \({\mathbf A}=\langle A, T, \leq\rangle\), where \(\langle A, \leq\rangle\) is a partially ordered set and \(\langle A,T\rangle\) is an algebra whose language \(T\) is a finite set consisting of a set \(T_0\) of constants, sets \(T_1\) and \(T_2\) of unary and binary residuated operations, respectively, and the sets of their residuals, \(T_1^*\) and \(T_2^*\), respectively. Hence \(T = T_0 \cup T_1 \cup T_2 \cup T_1^* \cup T_2^*\). For instance, a commutative residuated lattice \((A, \wedge, \vee, \circ, \rightarrow,1)\) is such an roa with \(T_0=\{1\}\), \(T_1= T_1^*= \emptyset\), \(T_2=\{ \circ \}\), \(T_2^*=\{\rightarrow \}\), with the partial order \(\leq\) being a lattice order with lattice operations \(\wedge\) and \(\vee\). The paper mainly presents the construction of a structure \({\mathbf F}=\langle F, T^F, \leq^F\rangle\), starting with a fixed roa \({\mathbf A}=\langle A, T, \leq\rangle\), a fixed partial (ordered) subalgebra \({\mathbf B}=\langle B, T^B, \leq^B\rangle\) of \({\mathbf A}\) and a fixed \({\mathbf B}\)-residual pair \({\mathbf M}=\langle M,M^*\rangle\), where \(F\) is the set of all stable sets \(X \subseteq M^*\) (i.e., verifying \(X^{lu}=X\)). The main result is Theorem 5.11, saying that \({\mathbf F}\) is a complete roa of the same type as \({\mathbf A}\), and there exists an embedding of \({\mathbf B}\) into \({\mathbf F}\) preserving all existing meets and joins in \({\mathbf B}\). The author then studies: which properties of \({\mathbf A}\) are preserved by \({\mathbf F}\); the conditions under which \({\mathbf F}\) is finite; the class of residuated ordered semigroups in which \(T_1=\emptyset\); the class of descending roas; the class of unary roas; the class of involutive residuated ordered semigroups; the class of roas satisfying an inequality \(s \leq t\). Finally, he proves that for commutative residuated lattices, the structure \({\mathbf F}\) and the structure \({\mathbf D}\) (constructed in two previous joint papers by Blok and the author) are isomorphic. Reviewer's remark: Since commutative residuated lattices (of the form \({\mathbf A}=(A, \wedge, \vee, \circ, \rightarrow,1)\)) are categorically equivalent to BCK-lattices with product (of the form \({\mathbf A}^*=(A, \wedge, \vee, \rightarrow, \circ, 1)\)), we think that a `dual' construction may be made, starting with `dual' roas (of the form \({\mathbf A^*}=\langle A, T^*, \leq \rangle\)), categorically equivalent to roas (of the form \({\mathbf A}\)), starting from residua (\(T^*=T_0 \cup T_1^* \cup T_2^* \cup T_1 \cup T_2=T\)), and considering \(F^*\) as the set of `dual' stable sets \(X \subseteq M\) (i.e., verifying \(X^{ul}=X\)). We expect that the `dual' structure \({\mathbf F^*}=\langle F^*, T^{F^*}, \leq^{F^*}\rangle\) thus obtained will be a complete `dual' roa, categorically equivalent to \({\mathbf F}\) (by a `dual' of Theorem 5.11).
scientific article; zbMATH DE number 5782716