On a theorem of Cantor-Bernstein type for algebras

Cantor–Bernstein Theorem for Pseudo-BCK-Algebras

“…(This is also a corollary of our Lemma 3.1.) Therefore, by Example 2.2(b) and Theorem 4.3 we gain: This result is due to Jakubík [10] and extends the following MV-algebraic CantorBernstein theorem which was proved by De Simone, Mundici and Navara [15] (it suffices to observe that every σ -complete MV-algebra is automatically an orthogonally σ -complete pseudo-MV-algebra):…”

The Cantor–Bernstein–Schröder theorem via universal algebra

“…However, there are L-varieties where the σ-completeness (orthogonally σ-completeness) condition on the algebras guarantee the corresponding σ-completeness (orthogonally σ-completeness) of their centers and then, the CBS F C -property. Examples of these particular L-varieties are: Boolean algebras (where the CBS F Cproperty was obtained by Sikorski and Tarski), Orthomodular lattices (where the CBS F C -property was obtained in [11]), MV-algebras (where the CBS F Cproperty was obtained in [10]), Pseudo MV-algebra (where the CBS F C -property was obtained in [22] Example 4.6 [Semigroups with 0, 1 and bounded semilattices] A semigroup with 0, 1 is an algebra A, •, 0, 1 of type 2, 0, 0 such that the operation • is associative, 0 • x = x • 0 = 0 and 1 • x = x • 1 = x. Thus, semigroups with 0, 1 define a variety denoted by SG 0,1 .…”

“…Following this idea, several authors have extended the Sikorski-Tarski version to classes of algebras more general than Boolean algebras. Among these classes there are lattice ordered groups [23], M V -algebras [10,21], orthomodular lattices [11], effect algebras [24], pseudo effect algebras [13], pseudo M V -algebras [22], pseudo BCK-algebras [30] and in general, algebras with an underlying lattice structure such that the central elements of this lattice determine a direct decomposition of the algebra [15]. It suggests that the CBS-theorem can be formulated in a common algebraic framework from which all the versions of the theorem mentioned above stem.…”

On complete pseudo-BL-algebras. A Cantor-Bernstein type theorem

“…The theorem was extended to σ-complete Boolean algebras by Sikorski [21] and Tarski [22]. Some theorems of Cantor-Bernstein type for (pseudo-) MV-algebras and for pseudo-effect algebras were proved in [15], [4], [16] and [7]. In the final section of this paper we generalize the Cantor-Bernstein theorem to σ-dually Brouwerian pseudo-BL-algebras.…”