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Elementary properties of free extensions

Authors:	William A Lampe Dale Myers

Institution:	(1) University of Hawaii, Honolulu, Hawaii, USA

Abstract:	For any partial groupoid $\mathfrak{A}$ , let Fr $\mathfrak{A}$ be the free extension of $\mathfrak{A}$ to a total groupoid. We show that $\mathfrak{A} \equiv \mathfrak{B}$ implies $Fr(\mathfrak{A}) \equiv Fr(\mathfrak{B})$ and that the theory of Fr $\mathfrak{A}$ is uniformly recursive in the theory of $\mathfrak{A}$ . These results fail if “groupoid” is replaced by “semigroup”, “commutative semigroup”, “group”, “abelian group”, “semilattice”, “K-lattice” for any nontrivial varietyK of lattices, or “Boolean algebra”. Research supported in part by NSF Grant MCS78-01867. We thank the referee for his valuable comments. Presented by B. Jónsson.

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