A note on atoms in polymodal algebras (Q1272114)

By an \(n\)-modal algebra is meant an algebra \(A=(A;\cap ,-,0,1\), \(\{\square _i; 0 \leq i \leq n\})\) such that \((A;\cap ,-,0,1)\) is a Boolean algebra and \(\square _i1=1\) and \(\square _i(a\cap b) = \square _ia\cap \square _i b\) for \(0\leq i \leq n\). A variety \({\mathcal V}\) of \(n\)-modal algebras is \(m\)-transitive if \(\square ^{(m)} a : = \bigcap \{\square ^i a; i \leq m\}\leq \square ^m a\) holds in \({\mathcal V}\), where \(\square ^0 a = a\) and \(\square ^{k+1} a = \bigcap \{\square _i \square ^k a; k \leq n\}\). \({\mathcal V}\) is weakly transitive if \({\mathcal V}\) is \(m\)-transitive for some integer \(m\). The main results: {Theorem 4}. Let \({\mathcal V}\) be a weakly transitive variety of modal algebras generated by its finite members. Then the free algebra with \(k\) free generators is atomic for every \(k\in N\). Denote by \(F(a)=\{b\in A; a \leq \square ^{(m)}b\) for all \(m\in N\}\). {Theorem 5}. Let \({\mathcal V}\) be a variety of modal algebras and \(a\) be an atom in the free algebra of \({\mathcal V}\). Then \(\square A \vDash F_{\mathcal V} (k)/F(a)\) splits \({\mathcal V}\).