Involutive Uninorm Logic with Fixed Point enjoys finite strong standard completeness

Abstract

An algebraic proof is presented for the finite strong standard completeness of the Involutive Uninorm Logic with Fixed Point ($${{\mathbf {IUL}}^{fp}}$$ IUL fp ). It may provide a first step towards settling the standard completeness problem for the Involutive Uninorm Logic ($${\mathbf {IUL}}$$ IUL , posed in G. Metcalfe, F. Montagna. (J Symb Log 72:834–864, 2007)) in an algebraic manner. The result is proved via an embedding theorem which is based on the structural description of the class of odd involutive FL$$_e$$ e -chains which have finitely many positive idempotent elements.

Group Representation for Even and Odd Involutive Commutative Residuated Chains

“…Such algebras are finitely generated, and a usual way to prove finite strong standard completeness is to embed finitely generated FL e -chains into standard ones. In fact, the finite iteration in the representation theorem guided the definition of a finite sequence of consecutive embeddings in [9]. The strong standard completeness of the same logic via embedding arbitrary (not only finitely generated) countable chains into standard ones in the present paper is not only a stronger result but its proof is substantially simpler, too.…”

“…By leveraging this redefined theorem, it is demonstrated that both the variety of semilinear odd involutive FLe-algebras and its idempotent symmetric subvariety admits densification. Ultimately, employing the algebraic techniques introduced in [11], the proof of the strong standard completeness of Involutive Uninorm Logic with Fixed Point (IUL f p ) is established, thus strengthening the main result of [9].…”

“…As a direct application, we prove the densification property for two varieties of odd involutive FL e -algebras, namely for the semilinear one and its idempotent symmetric subvariety. Ultimately, we prove the strong standard completeness of Involutive Uninorm Logic with Fixed Point (IUL f p ) in the algebraic style of [11], thereby surpassing the main result of [9].…”

“…As said in the introduction, strong standard completeness of a logic L can be shown by proving that any countable L-chain is embeddable into a standard Lchain, and that it can be achieved by densification and a subsequent MacNeille completion. Involutive Uninorm Logic with Fixed Point (IUL f p ) has been introduced in [16], we refer the reader to the introduction of [9] for a motivation for this interesting logic. Knowing that IUL fp -chains, that is, non-trivial bounded odd involutive FL e -chains constitute an algebraic semantics of IUL f p , we shall prove in this paper that any non-trivial countable, bounded odd involutive FL e -chain embeds into an odd involutive FL e -chain over the real unit interval [0, 1] such that its top element is mapped into 1 and its bottom element is mapped into 0.…”

“…To exhibit one more application, we prove here the strong standard completeness of IUL f p . The finite strong standard completeness of this logic has already been settled in [9]. Its proof relied on a representation theorem of those odd involutive FL e -chains which has only finitely many positive idempotent elements [11,8], and that representation theorem makes use of finite iteration of the so-called partial sublex product construction.…”