Ockham algebras arising from monoids (Q2762292)

An algebra \(\mathcal L=(L,\vee ,\wedge ,0,1,f)\) of type (2,2,0,0,1) is an Ockham algebra if \((L,\vee ,\wedge ,0,1)\) is a distributive \((0,1)\)-lattice and \(f\) is its dual endomorphism. If \((L,\vee ,\wedge ,0,1)\) is a Boolean lattice and \(f\) is not a complementation, then we say that \(\mathcal L\) is of Boolean shape, if \((L,\vee ,\wedge ,0,1)\) is a Boolean lattice and \(f\) is a bijection, then we say that \( \mathcal L\) is of Boolean type. For a monoid \(M\) and \(c\in M\), define \(f_c(A)=\{x\in M;xc\in M\setminus A\}\) for any \(A\subseteq M\). It is proved that \(\mathcal L^M_c=(2^M,\cup ,\cap ,\emptyset ,M,f_c)\) is an Ockham algebra such that if \(c\neq 1\), then \(\mathcal L^M_c\) is of Boolean shape, and if \(c\) is a unit of \(M\), then \(\mathcal L^M_c\) is of Boolean type. The authors characterize when \(\mathcal L^M_c\) belongs to the Urquhart class \(\mathbf P_{m,n}\), or \( \mathcal L^M_c\) belongs to the Berman class \(\mathbf K_{p,q}\), or \(\mathcal L^M_c\) is simple, or \(\mathcal L^M_c\) is subdirectly irreducible. If \(c\) belongs to the center of \(M\), then \(M\) is a subsemigroup of the endomorphism monoid of \(\mathcal L^M_c\). Hence any Urquhart class \(\mathbf P_{m,n}\) contains for any semigroup \(S\) the algebra \(\mathcal A\) such that \(S\) is a subsemigroup of the endomorphism monoid of \(\mathcal A\). A stronger version of this fact was proved for Kleene algebras in a paper of \textit{M. E. Adams} and \textit{H. A. Priestley} [``Kleene algebras are almost universal'', Bull. Aust. Math. Soc. 34, 343-373 (1986; Zbl 0583.06009)].