Small orthomodular partial algebras. (Q2343035)

The present paper deals with orthomodular partial algebras (OMAs). There exists a natural bijection between OMAs and orthomodular posets. An OMA is said to be Boolean if the corresponding poset is a Boolean lattice. OMAs with finite blocks are representable by means of Greechie diagrams. The authors prove necessary and sufficient conditions under which an OMA is strongly embeddable into a Boolean OMA. Next, the paper contains a full list (Greechie diagrams) of OMAs possessing at most 24 elements; the validity of the description was verified by hand and by a computer program, too. Particularly, the number of them is 121, 9 are not strongly embeddable into a Boolean OMA and 26 fail to have a lattice order. Finally, an infinite OMA with 4 generators is described by means of recursion; here the degree of any element is at most 4, on the contrary to the infinite 4-generated OMA described by \textit{V. Rogalewicz} [Math. Slovaca 41, No. 4, 431-435 (1991; Zbl 0771.03023)], where the degree was not bounded.