(Here I'm stepping on dangerous territory since many fields in mathematics have different ideas about what models are. Just so you know I'm writing this from an algebraic logic/model theory perspective. So if you can describe other types of models node them! Suggested reading for this definition includes formal languages, interpretation functions and model theory.)

First we start with a formal language, this formal language will contain some of the following: variables, functional symbols F and relational symbols R (both F and R are sets). Secondly we need a set, labelled S which is called the universe of our model. Lastly we need an interpretation function labelled I that interprets our language as subsets of S. A model M is then a triple <S, FI, RI> where FI contains interpretations or all the functional symbols and RI contains interpretations of the relational symbols.

If there aren't any functional symbols, i.e. F is empty, then we call M a relational structure or a relational model. On the otherhand if there aren't any relational symbols we call M an algebra. (This last part might make group theorists cry out in exasperation, but hey I'm an algebraic logician and who cares about their old fashioned ideas anyway?)

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