Nuclei and conuclei on residuated lattices
The authors prove some basic properties of nuclei and conuclei on residuated lattices. For a residuated lattice \(L = (L, \wedge, \vee, \otimes, \searrow, \swarrow, e)\), a map \(j : L\to L\) is called a nucleus if it is a closure operator and moreover satisfies \(j(a)\otimes j(b) \leq j(a\times b)\). It is proved, for example: {\parindent=6,5mm \begin{itemize}\item[(1)] A residuated lattice \(L\) is a GMV-algebra if and only if \(L_{w_{rl}^a}\) and \(L_{w_{lr}^a}\) are filters in \(L\) for all \(a\in L\). (Theorem 2.7) \item[(2)] If \(j\) is a nucleus on \(L\), then \(j\leq w_{rl}^c\) (or \(j \leq w_{rl}^c\)) if and only if \(j(c)=c\). (Proposition 2.8) \end{itemize}} Here \(w_{rl}^a (x) = a\swarrow (x \searrow a)\) and \(w_{lr}^a (x) = (a\swarrow x) \searrow a\) are closure operators on \(L\), and \(L_j = \{ x\in L\mid j(x) = x \}\) for a map \(j\). The authors also define a pseudo-dual quantale \(Q= (Q, \otimes, {}^\top, {}^\perp)\) which is a quantale \((Q,\otimes)\) with two order-reversing operations \({}^\top, {}^\perp\) satisfying CN1 and CN2: \[ \text{CN1:}\quad (a^\top)^\perp= a= (a^\perp)^\top, \qquad \text{CN2:}\quad a\otimes b\leq c \iff b\otimes c^\top\leq a^\top. \] Many results about pseudo-dual quantales are proved; for example: {\parindent=6,5mm \begin{itemize}\item[(3)] If \(Q\) is a dual quantale then it is also a pseudo-dual quantale. (Proposition 3.20) \item[(4)] A pseudo-dual quantale \(Q\) is a dual quantale if and only if it is a unital quantale. (Theorem 3.21) \item[(5)] A pseudo-Girard quantale \(Q\) is a Girard quantale if and only if it is a unital quantale. (Theorem 3.27) \end{itemize}} A pseudo-dual quantale \(Q\) is called a pseudo-Girard quantale if it satisfies \((a)^\top= (a)^\perp\) for all \(a\in Q\).
scientific article; zbMATH DE number 5924622
scientific article