LATTICES OF FREE EXTENSIONS IN CATEGORIES OF PARTIALLY ALGEBRAIC STRUCTURES

Grzegorz Jarzembski
1987 Demonstratio Mathematica  
It is not a big risk to say that the class of surjective epimorphisns is big enough if we dfeal with classical problems of the theory of total algebras. None of the basio concepts and fundamental theorems needs others then surjeotive epis. Only in some special varieties non-surjeotive epis are considered -for example, localizations in the theory of rings. In our .opinion among all reasons of it the most important are the following, rirst -each •pi in a category of all algebras of a given type
more » ... s of a given type is surjective. Second -each algebra in a given variety V is a surjective image of a suitable free V-algebra. And third -in any variety there exists a factorization system (Surjectiona, Monos). But working with partial algebras we can not restriot our attention to surjective epis only. One can easily oheok that none of these three postulates remains true for surjections in the theory of partial algebras. Hence we propose to distinguish another class of epis in categories of partially algebraic structures which may play the same role as surjeotions plaQr for total algebras. More precisely; in each variety V of partial algebras we distinguish a composition class By • -Clepiy'Exty, where Clepiy is a class of all closed [2] This paper is based on the lecture presented at the Conference on Universal Algebra held at
doi:10.1515/dema-1987-1-215 fatcat:sbxuerzmojdh5ibbnd5ppiffpm