Left residuated lattices induced by lattices with a unary operation
The authors recently have shown that every orthomodular lattice \(L\) can be converted into a left residuated lattice with operations \(\otimes\) and \(\to\) defined as term operations \((x \vee y') \wedge y\), resp., \((x \wedge y) \vee x'\) in \(L\); see [the authors, Miskolc Math. Notes 18, No. 2, 685--689 (2017; Zbl 1399.06020)]. This result is now generalized. Let now \(L\) be a bounded lattice with an unary operation \('\) and binary operations \(\otimes\) and \(\to\) defined as above; then this lattice is left residuated and satisfies the equation \(x \to 0 = x\) if and only if the identities \(x \wedge (((x \vee y') \wedge y) \vee y') = x\) and \(x \vee (((x \wedge y) \vee y') \wedge y) = x\) are valid in \(L\). In the rest of the paper, the connections between these two identities and weak and dually weak orthomodularity in the sense of [the authors, Asian-Eur. J. Math. 11, No. 2, Article ID 1850093, 18 p. (2018; Zbl 1386.06009)] are investigated. Finally, the mentioned result is improved: with the simple definitions \(x \otimes y = x \wedge y\) and \(x \to y = x' \wedge y\), the two identities may be replaced by \(x \wedge ((x \wedge y \vee y') = x\) and \(x \vee ((x \vee y') \wedge y) = x\).
scientific article; zbMATH DE number 7220820