Free ternary algebras (Q2709978)

A ternary algebra is an algebra \((L,+,\cdot, {}^\sim, 0,1,e)\) of type \((2,2,1,0,0,0)\) such that \((L,+,\cdot, 0,1)\) is a bounded distributive lattice and the following identities are satisfied: \((a+b)^\sim= a^\sim b^\sim\), \(a^{\sim\sim}= a\), \(e\leq a+a^\sim\), \(e^\sim= e\), \(0^\sim= 1\). Theorem 4.1 states that a set \(X\) of generators of a ternary algebra \(L\) freely generates \(L\) iff 1) \(\{0,e,1\}\cap X= \emptyset\), 2) \(X\cap X^\sim= \emptyset\), 3) if \(S\) and \(T\) are finite subsets of \(X\cup X^\sim\) and \(\Pi S\leq\Sigma T\) then (i) \(S\cap T= \emptyset\), or (ii) \(S\cap S^\sim\neq \emptyset\) and \(T\cap T^\sim\neq \emptyset\). It is shown that the free ternary algebra over one free generator consists of 11 elements (Corollary 4.1). The poset of all 18 join-irreducible elements of the free ternary algebra over two free generators is exposed.