In first-order logic there are the set of variables (that is considered as countable for whatever reason), terms, formulas, structures etc. Although that is said in metalanguage, is it possible to define naturally over these "sets" algebraic structures? E.g. over an alphabet the words with concatenation are associative and there is the empty word, therefore the set of words could be seen as a Monoid or not?
2026-03-30 13:37:20.1774877840
algebraic stuctures over metaconcepts in FO
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f you're talking arbitrary words from some alphabet (not necessarily well formed formulas or terms) under concatenation, then yes, this is the free monoid on your set of alphabetical characters. But more special classes of words can fail to be closed under concatenation.
A toy example of the latter case is the set of all words consisting only of repetitions of a single letter (so $x$ or $yyyyy$ but not $xyx$). Such strings are clearly not closed under concatenation. Generally speaking, another example is the WFFs of a first order language. $x\in yy=y$ is not itself, in most treatments of syntax that require some sort of propositional connective or something between formulas, a well formed formula.