The completions of multi-posets and quantum B-algebras

Abstract

Multi-posets as the generalization of quantum B-algebras and partially ordered semigroups etc. were firstly introduced and studied by Rump. In this paper, we shall continue his work and investigate the completions of multi-posets and quantum B-algebras. Firstly, we introduce the notions of the (precoherent) completions of a multi-poset M and the (algebraic) consistent quantic nuclei on the power-set $\mathfrak{P}(M^f)$ of the free semigroup of M, and prove that the (precoherent) completions of a multi-poset M is completely determined by the (algebraic) consistent quantic nuclei on $\mathfrak{P}(M^f)$ up to isomorphism. Moreover, we give the concrete forms of the least and the largest standard (precoherent) completions and obtain the free quantale over a multi-poset in the category of multi-posets. Finally, when the completions and consistent quantic nuclei of multi-posets are applied to the cases of quantum B-algebras and partially ordered semigroups, several analogous results can also be obtained.

Introduction

Quantales were introduced by Mulvey in order to provide a lattice-theoretic framework for studying non-commutative spaces, as well as a constructive foundation for quantum mechanics (see [12]). Quantale theory has become an active research topic in the field of mathematics, theoretical computer science and logic and have been applied extensively to logic, enriched category, quantum domain, lattice-valued topology and lattice-valued algebra, etc. ([13], [19], [20], [24], [25]).

Based on the implications of quantales and other implicational algebras, Rump and Yang introduced the concept of quantum B-algebras. It was shown that quantum B-algebras provide a unified semantic for a wide class of substructural logics. Just as Rump said: “every implicational algebra is contained in a quantum B-algebra would not be too far from the truth”, quantum B-algebras indeed cover the majority of implicational algebras like BCK-algebras, residuated lattices, residuated ordered algebras, partially ordered groups, BL- and MV-algebras, effect algebras, and their non-commutative extensions. In recent years, Rump, Yang, Botur, Paseka, Han and Zhang et al. have studied the quantum B-algebras from various aspects (see [3], [8], [9], [14], [15], [16], [17], [18], [26]). Especially, Rump and Han constructed the injective hulls in the category of quantum B-algebras respectively. As we know, the injective hulls in the category of quantum B-algebras are unique up to isomorphism, and hence the above structures are actually isomorphic. Furthermore, Rump introduced the generic structure of multi-posets that admits an essential embedding into a quantale. When applied to non-commutative topology, the multi-posets can lead to a symmetric version of quantum spaces as a class of spatial quantales. Furthermore, it was also proved that the category of promonoidal posets is equivalent to the category of unital coherent multi-posets.

Just as complete lattices and quantales can be viewed as the completions of posets and partially ordered semigroups respectively, quantales can also be treated as the completions of multi-posets and quantum B-algebras. It is well known that the join-completions of a poset are completely determined by the consistent closure operators up to isomorphism and the Dedekind MacNeille completion of a poset X is not only the least standard completion of X but also the injective hull of X in the category of posets (see [1], [2]). In order to describe the revised semantic for intuitionistic linear logic, Larchey-Wendling and Galmiche introduced the concept of quantale completions of partially ordered monoids in [11]. Furthermore, similar to the work of Larchey-Wendling and Galmiche, Han and Zhao considered the concept of quantale completions of partially ordered semigroups, and proved that the quantale completions of a partially ordered semigroup S are completely determined by the consistent quantic nuclei on $\mathfrak{P}(S)$ up to isomorphism (see [7]). The least and the largest consistent quantic nuclei on $\mathfrak{P}(S)$ were also given in [7] and [23]. In this paper, we shall investigate the completions and consistent quantic nuclei of multi-posets and show that the completions of a multi-poset M are completely determined by the consistent quantic nuclei on $\mathfrak{P}(M^f)$ up to isomorphism. Moreover, the concrete forms of the least and the largest standard completion of the multi-poset M are constructed, where the least standard completion of M is just the completion defined in [16] and the largest standard completion of M is the collection of so-called m-lower-sets of M. Finally, the related results for quantum B-algebras and partially ordered semigroups are also obtained.