In this paper, we continue the study of injectivity for fuzzy-like structures. We extend the results of Zhang and Laan for partially ordered semigroups to the setting of S-semigroups. We first ...characterize injectives in the category Ssgr⩽ of S-semigroups with subhomomorphisms as S-semigroup quantales. Second, we show that every S-semigroup has an E⩽-injective hull, and give its concrete form. Third, connections to ordered semicategories and quantaloids are indicated. In particular, if S is a commutative quantale, then the injectives in the category of S-semigroups with subhomomorphisms generalize the quantale algebras introduced by Solovyov. Quantale algebras provide a convenient universally algebraic framework for developing lattice-valued analogues of fuzzification.
In this paper, we continue the study of injectivity for fuzzy-like structures. We extend the results of Zhang and Paseka for
S
-semigroups to the setting of residuated
S
-posets. It turns out that ...every residuated
S
-poset over a quantale
S
embeds into its MacNeille completion as its
E
≤
-injective hull. In particular, if
S
is a commutative quantale, then the injectives in the category of residuated
S
-posets with subhomomorphisms are precisely the quantale algebras introduced by Solovyov. Quantale algebras provide a convenient universally algebraic framework for developing lattice-valued analogues of fuzzification.
This paper is devoted to the study of injectivity for ordered universal algebras. We first characterize injectives in the category
of ordered
-algebras with lax morphisms as sup-
-algebras. Second, ...we show that every ordered
-algebra has an
-injective hull, and give its concrete form.
QUANTALE-VALUED SUP-ALGEBRAS Slesinger, Radek
Iranian journal of fuzzy systems (Online),
03/2018, Letnik:
15, Številka:
2
Journal Article
Recenzirano
Odprti dostop
Based on the notion of $Q$-sup-lattices (a fuzzy counterpart of complete join-semilattices valuated in a commutative quantale), we present the concept of $Q$-sup-algebras -- $Q$-sup-lattices endowed ...with a collection of finitary operations compatible with the fuzzy joins. Similarly to the crisp case investigated in \cite{zhang-laan}, we characterize their subalgebras and quotients, and following \cite{solovyov-qa}, we show that the category of $Q$-sup-algebras is isomorphic to a certain subcategory of a category of $Q$-modules.
A 2-form between two sup-lattices
L and
R is defined to be a sup-lattice bimorphism
L×
R→2. Such 2-forms are equivalent to Galois connections, and we study them and their relation to quantales, ...involutive quantales and quantale modules. As examples we describe applications to C∗-algebras.
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