On the strong amalgamation of Boolean algebras

We show that the variety of ortholattices has the strong amalgamation property and that the variety of orthomodular lattices has the strong Boolean amalgamation property, i.e. that two orthomodular lattices can be strongly amalgamated over a common Boolean subalgebra. We give examples to show that the variety orthomodular lattices does not have the amalgamation property and that the variety of modular ortholattices does not even have the Boolean amalgamation property. We further show that no non-Boolean variety of orthomodular lattices which is generated by orthomodular lattices of bounded height can have the Boolean amalgamation property.

For any amalgam ( S , T ; U ) (S,\,T;\,U) of inverse semigroups, it is shown that the natural partial order on S ∗ U T S{{\ast }_U}T , the (inverse semigroup) free product of S S and T T amalgamating U U , has a simple form on S ∪ T S \cup T . In particular, it follows that the semilattice of S ∗ U T S{{\ast }_U}T is a bundled semilattice of the corresponding semilattice amalgam ( E ( S ) , E ( T ) ; E ( U ) ) (E(S),\,E(T);\,E(U)) ; taken jointly with a result of Teruo Imaoka, this gives that the class of generalized inverse semigroups has the strong amalgamation property. Preserving finiteness is also considered.

POLYNOMIAL NORMAL FORMS AND THE EMBEDDING OF POLYNOMIAL ALGEBRAS

We prove the preservation theorems for NATP; many of them extend the previously established preservation results for other model-theoretic tree properties. Using them, we also furnish proper examples of NATP theories which are simultaneously TP2 and SOP. First, we show that NATP is preserved by the parametrization and sum of the theories of Fraïssé limits of Fraïssé classes satisfying strong amalgamation property. Second, the preservation of NATP for two kinds of dense/co-dense expansions, i.e. the theories of lovely pairs and of H-structures for geometric theories, and dense/co-dense expansion on vector spaces is proved. Next, we prove the preservation of NATP for the generic predicate expansion and the pair of an algebraically closed field and its distinguished subfield; for the latter, not only NATP, but also the preservation of NTP1 and NTP2 is considered. Finally, we present some proper examples of NATP theories using the results proved in this paper, including the parametrization of DLO and the expansion of an algebraically closed field by adding a generic linear order. In particular, we show that the model companion of the theory of algebraically closed fields with circular orders (ACFO) is NATP.