Pre-ideals of Basic Algebras

Abstract

Basic algebras are a generalization of MV-algebras, also including orthomodular lattices and lattice effect algebras. A pre-ideal of a basic algebra is a non-empty subset that is closed under the addition ⊕ and downwards closed with respect to the underlying order.In this paper, we study the pre-ideal lattices of algebras in a particular subclass of basic algebras which are closer to MV-algebras than basic algebras in general. We also prove that finite members of this subclass are exactly finite MV-algebras.

A note on derivations on basic algebras

“…It follows from Krňávek and Kühr (2011), Theorem 4.4, as well as from Botur and Kühr (2014), Theorem 4.7, that every finite basic algebra satisfying (2) is an MV-algebra. 1…”

“…which are equivalent to one another, are equivalent to lattice distributivity, i.e., a basic algebra fulfills the identities (1) if and only if its underlying lattice is distributive (see Krňávek and Kühr 2011). Finally, the identities…”

“…Basic algebras were introduced in Chajda et al (2009) as algebras corresponding to lattices with antitone involutions in the following natural way (also see Chajda and Kühr 2013a, b;Botur and Kühr 2014;Krňávek and Kühr 2011): (i) Given a bounded lattice (A, ∨, ∧, 0, 1) with antitone involutions (β a ) a∈A , let ¬x = x 0 and x ⊕ y = (x 0 ∨ y) y . Then (A, ⊕, ¬, 0) is a basic algebra from which the lattice can be retrieved by x ∨ y = ¬(¬x ⊕ y) ⊕ y, x ∧ y = ¬(¬x ∨ ¬y) and 1 = ¬0; the antitone involutions β a :…”

“…In order to define central elements, we first need to recall the following simple result (see Krňávek andKühr 2011 or Chajda andKolařík 2009b):…”

“…InKrňávek and Kühr (2011), this result was proved for basic algebras satisfying the identity x ⊕ (¬x ∧ y) = x ⊕ y, and inBotur and Kühr (2014), for those satisfying the identity x ≤ x ⊕ y. Note that the former identity holds in all linearly ordered basic algebras.…”