Principal congruences on \(S_1\)-algebras (Q2854030)

An Ockham algebra \((L,f)\) is a bounded distributive lattice with a dual endomorphism \(f\). An \(S_1\)-algebra is any Ockham algebra \((L,f)\) fulfilling the identities \(f=f^3\) and \(f(x)\wedge f^{2}(x)=0\) [\textit{R. Beazer}, Glasg. Math. J. 25, 175--181 (1984; Zbl 0539.06012)].NEWLINENEWLINEFor an \(S_1\)-algebra \((L,f)\) the relation \(\Phi\) defined on \(L\) by \((x,y)\in \Phi\) iff \(f(x)=f(y)\) is a congruence, called the determination congruence of \(L\). For every distributive lattice \(L\) the symbol \(J(L)\) denotes the poset of join-irreducible elements of \(L\), the lenght \(l(P)\) of a poset \(P\) is the supremum of the lengths of its chains. The main aim of the paper is to desribe those \(S_1\)-algebras all conguences of which are principal (this property is denoted (PC)):NEWLINENEWLINETheorem. An \(S_1\)-algebra \((L,f)\) has the (PC) property iff the determination congruence \(\Phi\) is principal and \(l(J([0]\Phi))\leq 1\), \(l(J([1]\Phi))\leq 1\).