Intersection of finitely generated congruences over term algebra

We show that it is decidable for any given ground term rewrite systems R and S if there is a ground term rewrite system U such that ↔U* = ↔R* ∩ ↔S*. If the answer is yes, then we can effectively construct such a ground term rewrite system U. In other words, for any given finitely generated congruences ρ and τ over the term algebra, it is decidable if ρ ∩ τ is a finitely generated congruence. If the answer is yes, then we can effectively construct a ground term rewrite system U such that ↔U* = ρ ∩ τ.

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