Quasi-rings and congruences in the theory of orthomodular algebras

Abstract

In this paper the theory of orthomodular algebras (OMAs) is further developed. The connection with quasi-rings and the structure of the ordered sets of their congruence relations are investigated. The paper is divided into five sections: on the structure of OMAs, the extensions of orthomodular addition, orthomodular algebras and orthomodular quasi-rings, OMA congruences, and OMA-Schmidt congruences. In conclusion, all OMA-Schmidt congruence relations for very small OMAs are computed and the line diagram of the OMA-Schmidt congruence lattice for the OMA with two free generators is constructed.

On subdirectly irreducible OMAs

“…Let A be a Boolean OMA, then A ∈ ISP (2) holds. 9 see [BM98] and [BM94] Proof. In the signature of Boolean algebras the induced Boolean algebra (A, ∧, ∨, , 0, 1) is isomorphic to a subalgebra of a power of the algebra C 2 with two elements, so there exists an injective homomorphism f : (A, ∧, ∨, , 0, 1) → C I 2 for a set I.…”

“…In the signature of Boolean algebras the induced Boolean algebra (A, ∧, ∨, , 0, 1) is isomorphic to a subalgebra of a power of the algebra C 2 with two elements, so there exists an injective homomorphism f : (A, ∧, ∨, , 0, 1) → C I 2 for a set I. With [BM98] f is also a homomorphism in the signature of OMAs. Now we show that f is closed.…”

Greechie diagrams of orthomodular partial algebras

“…is an OMA-homomorphism between Boolean OMAs and therefore a Boolean lattice homomorphism (see [BM98]). Of course φ is surjective.…”

Algebraic theory of quasivarieties of heterogeneous partial algebras

“…in [30], the language would not have enough expressive power. 18 However, we can define in this note a "usual heterogeneous first order language with identity", since using the signature Σ e "we now do have enough terms". In particular, if X has the elements x 1 , .…”

“…In [17], [18] and [27] the ECE-variety of orthomodular partial algebras 7 has been investigated. And in many papers and a book [20] by S.Pulmannova and A.Dvurečenskij some more quasivarieties of partial algebras, among others those of effect algebras and D-algebras, have been investigated.…”