Imaginaries in Boolean algebras

Given an infinite Boolean algebra B, we find a natural class of ∅-definable equivalence relations EB such that every imaginary element from B is interdefinable with an element from a sort determined by some equivalence relation from EB . It follows that B together with the family of sorts determined by EB admits elimination of imaginaries in a suitable multisorted language. The paper generalizes author’s earlier results concerning definable equivalence relations and weak elimination of imaginaries for Boolean algebras, obtained in [We3]. 0 Introduction and preliminaries Although no infinite Boolean algebra admits elimination of imaginaries, there exist infinite Boolean algebras admitting weak elimination of imaginaries. As proved in [We3], an infinite Boolean algebra B admits weak elimination of imaginaries iff the quotient Boolean algebra B/I(B) consists of at most two elements (here I(B) denotes the ideal of B consisting of all elements of the form atb with a atomless and b atomic). A special case of this result (namely: weak elimination of imaginaries for infinite Boolean algebras with finitely many atoms) plays a crucial role in studying definable sets of partially ordered o-minimal structures with ordering derived from a Boolean algebra. C. Toffalori in [To] introduced two notions of o-minimality for partially ordered first-order structures. A partially ordered structure M = (M,≤, . . .) is called quasi o-minimal if every definable set X ⊆ M is a finite Boolean combination of sets defined by inequalities of the form x ≤ a and x ≥ b, where a, b ∈ M . If additionally the parameters appearing in these inequalities may be taken from the algebraic closure of the set of parameters needed to define X, then the structure M = (M,≤, . . .) is called o-minimal. It is easy to see that in case the ordering ≤ is linear, these two notions are equivalent to the usual o-minimality. C. Toffalori observed that if M = (M,≤, . . .) is quasi o-minimal and the ordering ≤ comes from some Boolean algebra B, then the number of atoms of B must be finite. By weak elimination of imaginaries for Boolean algebras with finitely atoms, the Toffalori’s notions of o-minimality and quasi o-minimality coincide in case of Boolean ordered structures. A natural counterpart of o-minimality (called q-minimality) for expansions of arbitrary Boolean algebras was introduced in author’s PhD thesis. An expansion (B, . . .) of a Boolean algebra B to the language L ⊇ LBA is said to be q-minimal iff every L-definable subset of B is LBA-definable, where LBA = {u,t,′ , 0, 1} denotes the usual language of Boolean algebras. By results of [NW], for expansions of Boolean algebras with finitely many atoms, q-minimality coincides with Toffalori’s notions of quasi o-minimality and o-minimality. As the model theoretic results obtained in [NW], [We1] and [We2] for Boolean ordered structures heavily rest on weak elimination of imaginaries of the underlying Boolean algebras, it is natural to expect that some form of elimination of imaginaries will be needed to investigate sets definable in q-minimal expansions of arbitrary Boolean algebras. 0