STRONGLY MINIMAL PARTIAL ORDERINGS OF HEIGHT TWO

In the present paper, we study strongly minimal partial orderings in the signature containing only the symbol of binary relation of partial order. We use for partial orderings such characteristics as the height of a structure that is the supremum of lengths of ordered chains, and the width of a structure that is the supremum of lengths of antichains, where an antichain is a set of pairwise incomparable elements. We also differ trivial width and non-trivial width. Recently, B.Sh. Kulpeshov, In.I. Pavlyuk and S.V. Sudoplatov described strongly minimal partial orderings having a finite non-trivial width. Here we study strongly minimal partial orderings having an infinite non-trivial width. The main result of the paper is a criterion for strong minimality of an infinite partial ordering of height two having an infinite non-trivial width.

1. Baldwin J.T., Lachlan A.H. On strongly minimal sets. The Journal of Symbolic Logic, 1971, vol. 36, no. 1, pp. 79–96.

2. Kulpeshov B.Sh., Pavlyuk In.I., Sudoplatov S.V. On pseudo-strongly-minimal formulae, structures and theories. Model Theory and Algebra 2024, Collection of papers edited by M. Shahryari and S.V. Sudoplatov, Novosibirsk State Technical University, Novosibirsk, 2024, pp. 42–47.

3. Kulpeshov B.Sh., Pavlyuk In.I., Sudoplatov S.V. Pseudo-strongly-minimal structures and theories. Lobachevskii Journal of Mathematics, 2024, vol. 45, no. 12, pp. 6398–6408.

4. Kulpeshov B.Sh., Sudoplatov S.V. Properties of ranks for families of strongly minimal theories. Siberian Electronic Mathematical Reports, 2022, vol. 19, no. 1, pp. 120–124.

5. Baldwin J.T., Verbovskiy V.V. Towards a finer classification of strongly minimal sets. Annals of Pure and Applied Logic, 2024, vol. 175, no. 103376.