Consider the lattice of equational theories of a single binary operation $*$. The join of two finitely based equational theories is of course finitely based. Do there exist two finitely based equational theories whose meet is not finitely based?
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Meet means intersecting the equations and join means unioning them? Isn’t it the join that’s obviously finitely based? Or does “finitely based” not mean finitely axiomatizable? – Qiaochu Yuan Sep 13 '20 at 17:49
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@QiaochuYuan Hmm, I am not sure. I mean to ask whatever is not obviously finitely based, whether it be meet or join. – user107952 Sep 13 '20 at 18:24
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If "finitely based" means finitely axiomatizable then the join is finitely axiomatizable by the finite axioms of one theory + the finite axioms of another, yes? So it's the meet that's not obvious. – Qiaochu Yuan Sep 13 '20 at 18:26
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@QiaochuYuan I have edited the question. – user107952 Sep 13 '20 at 18:59
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1Admittedly the direction of the order is a bit tricky to pin down a convention for because the relationship between equations and models is contravariant - if we add more equations the theory gets bigger but the class of models gets smaller. – Qiaochu Yuan Sep 13 '20 at 22:37
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Such theories exist. I will describe a construction from the paper
Finitely Based, Finite Sets of Words.
M. Jackson, O. Sapir,
International Journal of Algebra and Computation 10(6):683-708 (2000).
Let $X=\{a,b\}$ and let $M(a,b)$ be the free monoid over $X$. If $W\subseteq X^*$ is a set of words in the letters $X$, let $I(W)$ be the ideal of $M(a,b)$ consisting of all non-identity monoid elements that are not subwords of words in $W$. Define $S(W) = M(a,b)/I(W)$.
In Theorem 5.8 of their paper, Jackson and Sapir show that
(i) $A:=S(\{abbaa, ababa, aabba\})$ is finitely based.
(ii) $B:=S(\{baaab, aabb, abba, abab\})$ is finitely based.
(iii) $A\times B$ is not finitely based.
This answers the question because $\textrm{Th}(A\times B)=\textrm{Th}(A)\cap \textrm{Th}(B)$.
Keith Kearnes
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