Varieties with directly decomposable subalgebras and homomorphisms (Q797611)

Let \({\mathcal V}\) be a variety. A semiconstant of \({\mathcal V}\) is a pair [c,d], where c,d are nullary operations of \({\mathcal V}\) (if such exist). By an n-ary semipolynomial of \({\mathcal V}\) is meant an object \(p([x^ 1_ 1,x^ 1_ 2],...,[x^ n_ 1,x^ n_ 2]),\) where p is an n-ary polynomial of \({\mathcal V}\) and any pair \([x^ i_ 1,x^ i_ 2]\), \(i=1,...,n,\) is either a pair of variables or a semiconstant of \({\mathcal V}\). \({\mathcal V}\) has directly decomposable subalgebras if for each \({\mathfrak A}_ 1,{\mathfrak A}_ 2\in {\mathcal V}\) and every semiconstant preserving subalgebra \({\mathfrak B}\) of \({\mathfrak A}_ 1\times {\mathfrak A}_ 2\) there exist subalgebras \({\mathfrak B}_ i\) of \({\mathfrak A}_ i\), \(i=1,2\), with \({\mathfrak B}={\mathfrak B}_ 1\times B_ 2\). \({\mathcal V}\) has directly decomposable homomorphisms if for each \({\mathfrak A}_ i,{\mathfrak B}_ i\in {\mathcal V}\), \(i=1,2\), and every semiconstant preserving homomorphism \(h:{\mathfrak A}_ 1\times {\mathfrak A}_ 2\to {\mathfrak B}_ 1\times {\mathfrak B}_ 2\) there exist homomorphisms \(h_ i:{\mathfrak A}_ i\to {\mathfrak B}_ i\), \(i=1,2\), with \(h=h_ 1\times h_ 2\). Theorem 1. \({\mathcal V}\) has directly decomposable subalgebras iff there exists a binary semipolynomial d of \({\mathcal V}\) such that \(d([x_ 1,x_ 2],[y_ 1,y_ 2])=[x_ 1,y_ 2]\). Theorem 2. If \({\mathcal V}\) has directly decomposable subalgebras, then it has directly decomposable homomorphisms. Examples of the varieties treated are included.