A Non-associative Generalization of Mv-algebras
نویسندگان
چکیده
We consider a non-associative generalization of MV-algebras. The underlying posets of our non-associative MV-algebras are not lattices, but they are related to so-called λ-lattices. c ©2007 Mathematical Institute Slovak Academy of Sciences 1. Non-associative MV-algebras As known, MV-algebras were introduced in the late-fifties by C . C . C h a n g as an algebraic semantics of the Lukasiewicz many-valued sentential logic (see [5], [6]). We recall the definition from [7] which is essentially due to P . M a n g a n i [12]; C h a n g ’s original definition in [5] was a bit more complicated: An MV-algebra is an algebra (A,⊕,¬, 0) of type (2, 1, 0) satisfying the following identities: (MV1) x⊕ (y ⊕ z) = (x⊕ y)⊕ z, (MV2) x⊕ y = y ⊕ x, (MV3) x⊕ 0 = x, (MV4) ¬¬x = x, (MV5) x⊕ ¬0 = ¬0 (the element ¬0 is denoted by 1), (MV6) ¬(¬x⊕ y)⊕ y = ¬(¬y ⊕ x)⊕ x. The prototypical example of an MV-algebra is the algebra Γ (G, u) = ([0, u],⊕,¬, 0), where (G,+,−, 0,∨,∧) is an Abelian lattice-ordered group, 0 < u ∈ G and [0, u] = {x ∈ G : 0 ≤ x ≤ u}, and the operations ⊕ and ¬ are defined via x⊕ y := (x+ y) ∧ u and ¬x := u− x, respectively. D . M u n d i c i 2000 Mathemat i c s Sub j e c t C l a s s i f i c a t i on: Primary 03G10, 06D35.
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