Endoprimal implication algebras (Q5932478)

Implication algebras are groupoids satisfying the identities \[ (x\circ y)\circ y =x,\quad (x\circ y)\circ y = (y\circ x)\circ x,\quad x\circ (y\circ z)= y\circ (x\circ z). \] An algebra \({\mathcal A} =(A,F)\) is \(n\)-endoprimal if every map \(f:A^n\to A\) that preserves the endomorphisms of \(\mathcal A\) is an \(n\)-ary term function of \(\mathcal A\). An algebra is endoprimal if it is \(n\)-endoprimal for all \(n\in N\). It is shown that there are no non-trivial endoprimal implication algebras. Infinite endoprimal implication algebras are built up from free implication algebras. The author gives a description of the \(n\)-endoprimal implication algebras for each \(n\in N\). If \(B\) is an infinite Boolean lattice then the join-semilattice on the set \(B \backslash \{ 0\}\) determines an endoprimal implication algebra.