Structures, homomorphisms, and the needs of model theory

Abstract

When we look closely at textbooks on model theory, we find that there are three different accounts of what a model or structure is. One of these is highly language dependent, so that the same structure cannot be the interpretation of two different languages or signatures. The other two definitions do not fall foul of that dependence but all textbooks tie the notion of homomorphism so closely to language (signature) that only structures interpreting the same language (signature) are isomorphic. Although this follows the practice in universal algebra, it is highly unnatural. The aim here is to present a notion of homomorphism better consonant with intuition and with what the less cautious authors of textbooks say when they speak informally.