On (3,3)-Homogeneous Greechie Orthomodular Posets

Abstract

We describe (3,3)-homogeneous orthomodular posets for some cardinality of their sets of atoms. We examine a state space and a set of two-valued states of such logics. Particular homogeneous OMPs with exactly k pure states (k = 1, . . . , 7, 10, 11) have been constructed.

Automorphisms of decompositions

“…✷ Remark 2.8. An omp is called (n, m)-homogeneous [28] if all of its blocks have m atoms and each atom is in n blocks. The above results show that if X is a set with p k elements for some prime p, then Fact X is (n, m)-homogeneous where m = k and n is given in Proposition 2.7.…”

“…For sets with p k elements for some prime p, and for finite-dimensional vector spaces over finite fields, these are seen to give interesting classes of (n, m)-homogeneous omps [28], that is, ones where each block has m atoms and each atom is in n blocks. Several less basic combinatorial properties of these structures are also considered, such as the relationship between Fact V when V is considered as a vector space and Fact V when V is considered as a set.…”

“…In this paper, the second section begins with some elementary combinatorial computations to find the number of atoms, number of blocks, size of blocks, and so forth, for the structures Fact X for a finite set X and Fact V for a finitedimensional vector space V . For sets with p k elements for some prime p, and for finite-dimensional vector spaces over finite fields, these are seen to give interesting classes of (n, m)-homogeneous omps [28], that is, ones where each block has m atoms and each atom is in n blocks. Several less basic combinatorial properties of these structures are also considered, such as the relationship between Fact V when V is considered as a vector space and Fact V when V is considered as a set.…”

Automorphism Groups of Small (3,3)-homogeneous Logics

“…The state spaces of some (3,3)-hom. logics of the form H k (m) (with cardA = m and k pure states) is studied in [3]. L 1 (19)).…”