On the definability of Lesniewski's copula 'is' in some ontology-like theories

We formulate a certain subtheory of Ishimoto's [1] quantifier-free fragment of Lesniewski's ontology, and show that Ishimoto's theory can be reconstructed in it. Using an epimorphism theorem we prove that our theory is complete with respect to a suitable set-theoretic interpretation. Furthermore, we introduce the name constant 1 (which corresponds to the universal name 'object') and we prove its adequacy with respect to the set-theoretic interpretation (again using an epimorphism theorem). Ishimoto's theory enriched by the constant 1 is also reconstructed in our formalism with into which 1 has been introduced. Finally we examine for both our theories their quantifier extensions and their connections with Le´sniewski's classical quantified ontology.