Decomposition of P{\l}onka sums into direct systems

Abstract

The P{\l}onka sum is an algebra determined using a structure called a direct system. By a direct system, we mean an indexed family of algebras with disjoint universes whose indexes form a join-semilattice s.t. if two indexes are in a partial order relation, then there is a homomorphism from the algebra of the first index to the algebra of the second index. The sum of the sets of the direct system determines the universe of the P{\l}onka sum. Therefore, to speak about a P{\l}onka sum, there must be a direct system on which this algebra is based. However, we can look at a P{\l}onka sum the other way around, considering it as some given algebra, and ask whether it is possible to determine all direct systems of it systematically. In our paper, we will decompose the P{\l}onka sum in such a way as to give a solution to the indicated problem. Moreover, our method works for any algebra of the kind considered in the article, and thus, we can determine if a given algebra is a P{\l}onka sum. The proposed method is based on two concepts introduced in the paper: isolated algebra and P{\l}onka homomorphism.