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Universal algebra has syntax and semantics parts.

  • A signature $\Sigma$ belongs to syntax.

  • Does $T(\Sigma,V)$, the set of terms for signature $\Sigma$ and a set $V$ of variables, belong to syntax or semantics?

In Baader's Term Rewriting Systems book, the beginning of Chapter 3 says

The purpose of this chapter is twofold. On the one hand, it introduces basic notions from universal algebra (such as terms, substitutions, and identities) on a syntactic level that does not require (or give) much mathematical background. On the other hand, it presents the semantic counterparts of these syntactic notions (such as algebras, homomorphisms, and equational classes), and proves some elementary results on their connections.

Most of the definitions and results presented in subsequent chapters can be understood knowing only the syntactic level introduced in Section 3.1. In order to obtain a deeper understanding of the meaning of these results, and of the context in which they are of interest, a study of the other sections in this chapter is recommended, however.

  • In Section 3.1, signature $\Sigma$ and $T(\Sigma,V)$ are introduced. So I thought $T(\Sigma,V)$ belongs to syntax.

  • In Section 3.2, for a given signature $\Sigma$, a $\Sigma$-algebra provides an interpretation of all the function symbols in $\Sigma$.

  • In Section 3.4, Term algebra $\mathcal{T}(\Sigma,X)$, where $X$ is a set of variables (not necessarily $V$) and the carrier is $T(\Sigma,V)$ ,is introduced. So I began to doubt whether $T(\Sigma,V)$ belongs to syntax.

Tim
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1 Answers1

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The set of terms generated by a signature is a syntactic notion (intuitively, because you could write out each of the elements by inductively building them from the operations and variables). However, the set of terms plays a special role in that it also forms a canonical model, or algebra: the term algebra. So, here, $T(\Sigma, X)$ is playing a special role as a syntactic structure that is equipped with canonical semantic structure given by the formation rules for the operations (in category theoretic language, it forms the initial model). (Strictly speaking, it is syntactic specifically when $X = \emptyset$, because arbitrary sets are not necessarily syntactic, but we can always treat an arbitrary set as being isomorphic to a canonical set of variables with the same cardinality.)

varkor
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