EXAMPLES OF O-MINIMAL EXPANSIONS OF THE DENSE MEET TREE

The notion of o-minimality is highly productive for linearly ordered structures, but a direct transfer of this concept to partially ordered sets encounters certain difficulties. Indeed, there is a striking scarcity of works on o-minimal partially ordered structures. The standard definition of o-minimality for partially ordered structures states that every definable subset is a Boolean combination of intervals and points. However, since Boolean combinations involve the operation of taking set complements, and in partially ordered sets the complement of an interval can be extremely complex, this approach presents certain challenges for studying the resulting class of structures. We propose using an alternative definition of o-minimality: a partially ordered structure is o-minimal if every definable subset is a finite union of generalized intervals and points. In the paper, we provide a number of examples demonstrating that this definition is nontrivial and that there exist structures which are o-minimal in this new sense.

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