Balanced pseudocomplemented Ockham algebras (Q2474104)

The authors investigate a particular subclass of pseudocomplemented Ockham algebras\break \((L,\wedge, \vee,f, *,0,1)\) where \((L,\wedge, \vee,f,0,1)\) is an Ockham algebra, \((L,\wedge, \vee,*,0,1)\) is a \(p\)-algebra, and the operations \(x\rightarrow f(x)\) and \(x\rightarrow x^{*}\) satisfy the identities \(f(x^{*})=x^{**}\) and \([f(x)]^{*}=f^{2}(x)\). An Ockham algebra is a bounded distributive lattice \(L\) together with a dual endomorphism \(f:L\rightarrow L\). The class of Ockham algebras is denoted by \(\mathbf{O}\). The well-known subclasses of Ockham algebras are Berman subclasses \(\mathbf{K}_{p,q}\) in which the dual endomorphism \(f\) satisfies the condition \(f^{2p+q}=f^{q}\), for \(p\geq 1\) and \(q\geq 0\). A double \(\mathbf{K}_{p,q}\)-algebra is an algebra \((L,f,k)\) in which both \((L,f)\) and \((L,k)\) are \(\mathbf{K}_{p,q}\)-algebras with a certain positive integer \(n\) such that \(fk=k^{2n}\) and \(kf=f^{2n}\). A (distributive) \(p\)-algebra (or lattice with pseudocomplementation) is a (distributive) lattice \(L\) with a smallest element 0 together with a mapping \(*:L\rightarrow L\) such that \(x\wedge y=0 \Leftrightarrow y\leq x^{*}\). In particular, a distributive \(p\)-algebra \((L,*)\) is called a Stone algebra if for every \(x\in L, x^{*}\vee x^{**}=1\). The class of distributive \(p\)-algebras is denoted by \(\mathbf{p}\), and that of Stone algebras by \(\mathbf{S}\). The authors consider a particular subvariety of pseudocomplemented algebras: a balanced pseudocomplemented Ockham algebra \((L,f,*)\) which is a bounded distributive lattice with two unary operations such that \((L,f)\in \mathbf{O}, (L,*)\in \mathbf{p}\) and \(f(x^{*})=x^{**},[f(x)]^{*}=f^{2}(x)\), for every \(x\in L\). The variety of balanced pseudocomplemented Ockham algebras is denoted by \(\mathbf{bpO}\). The authors show that there are precisely eleven non-isomorphic subdirectly irreducible members in the class of these algebras and give a complete description of them (by Hasse diagrams) in Theorem 12. These algebras form an ordered set under the following order: \(A\leq B \Leftrightarrow A\) is an isomorphic image of a subalgebra of \(B\).