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Let $A$ and $B$ be algebras of the same type such that the set of all their subdirect products consists of their ordinary product alone. Is there any terminology that describes this situation? I mean something like "we say that $A$ and $B$ are disjoint/orthogonal/...".

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  • $\begingroup$ This seems like an interest notion. Do you have an examples of this? $\endgroup$ Jul 8, 2016 at 14:47
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    $\begingroup$ I have not heard of a name for this. Is this equivalent to A, B, and AxB all being subdirectly irreducible? If you were building a matroid, you might use matroid terminology for this relationship. Gerhard "Hasn't Done This For Years" Paseman, 2016.07.08. $\endgroup$ Jul 8, 2016 at 15:29

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Libor Barto and Marcin Kozik invented (and subsequently proved many results about) a concept called "absorption," which doesn't exactly give the new terminology you asked for, but does give special conditions under which a subdirect product must be the full product.

In particular, they prove the main Absorption Theorem which states that (in a Taylor variety) if $A$ and $B$ are "absorption-free," then any "linked" subdirect product of $A\times B$ is the full product. I'm leaving out many details and all the definitions, but you can get them direct from the source. (See especially Theorem 2.3.)

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