Weak varieties of partial algebras (Q1112080)

The notion of weak variety of partial algebras, being a generalization of ECE-variety in the sense of P. Burmeister and \(H_ cS_ sP\)-variety in the sense of H. Andreka and I. Nemeti, is introduced and intensively studied. Namely, a nonempty class of partial algebras is called weak variety if it is closed under strong epimorphisms e \((f_ A(\underline x)\) is defined iff \(f_ B(e\underline x)\) is defined and \(ef_ A(\underline x)=f_ B(e\underline x))\) and also closed under strong monosources. A family \(\{f^ k: {\mathfrak A}\to {\mathfrak B}^ k\), \(k\in K\}\) of homomorphisms is called monosource iff it contains at least one strong homomorphism and for each pair of homomorphisms g,h: \({\mathfrak C}\to {\mathfrak A}\) the equality of \(f^ kg=f^ kh\) implies the equality \(g=h\). A characterization of weak varieties in the language of logic formulas is given. The problems under which conditions the variety in the above sense is closed under products or ultraproducts are studied.