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I'm reading Algebraic Theories by Ernest G. Manes,

and I'm wondering about equationally-definable class.

For example, group is a typical example of equationally-definable algebra,

but the definition of group uses ''for-all'' sentence: $\forall x,y,z\in X.\;x\cdot(y\cdot z)=(x\cdot y)\cdot z$;

thus I'm curious that whether can I use $\forall$, $\exists$, etc. in definition of equationally-definable algebra.

ぼっけなす
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  • As far as I know, "$\forall x \in X, \exists y \in Y$ such that..." is a very common mathematical sentence. – Marco Ripà Oct 09 '23 at 09:50
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    No. Equationally definable classes can only be axiomatized by universally quantified equations. – Emil Jeřábek Oct 09 '23 at 11:46
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    @MarcoRipà Yes, but "equationally-definable" is a very strict condition not allowing existential quantifiers (or even disjunctions). – Noah Schweber Oct 09 '23 at 15:54
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    Despite the presence of the words "algebraic" and "theory," the "algebraic-k-theory" tag is irrelevant; I've replaced it with a more relevant one. Separately, this question would be more appropriate at math.stackexchange. – Noah Schweber Oct 09 '23 at 15:55
  • @NoahSchweber Thank you! I'm sorry since I am new to mathoverflow and didn't find any relevant tag.... –  Oct 10 '23 at 01:50

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