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On the $$L$$-valued categories of $$L$$-$$E$$-ordered sets. (English) Zbl 1251.03062
There exists a well-known representation of a partially ordered set (poset) $$(X,\leqslant)$$ as a small category X whose objects are the elements of $$X$$ and whose hom-sets $$\text{hom}(x_1,x_2)$$ contain exactly one morphism provided that $$x_1\leqslant x_2$$; an order-preserving map $$f:(X,\leqslant)\rightarrow(Y,\leqslant)$$ is then precisely a functor $$F:\mathbf{X}\rightarrow \text\textbf{Y}$$ [J. Adámek, H. Herrlich and G. E. Strecker, Repr. Theory Appl. Categ. 2006, No. 17, 1–507 (2006; Zbl 1113.18001)], thereby providing the category Pos of posets and order-preserving maps.
Motivated by the notions of lattice-valued partial order of [L. A. Zadeh, Inf. Sci. 3, 177–200 (1971; Zbl 0218.02058)], the lattice-valued equality of [U. Höhle and N. Blanchard, Inf. Sci. 35, 133–144 (1985; Zbl 0576.06004)] (see also [U. Höhle and T. Kubiak, Fuzzy Sets Syst. 166, No. 1, 1–43 (2011; Zbl 1226.06011)] for the recent developments of the theory) and the lattice-valued category of [A. Šostak, Tatra Mt. Math. Publ. 16, No. 1, 159–185 (1999; Zbl 0947.18001)], the author of the current paper introduces a similar representation for the category $$\mathbf{Pos}(L)$$ of $$L$$-valued posets, where $$L$$ is supposed to be a commutative $$cl$$-monoid [U. Höhle, Theory Decis. Libr., Ser. B. 32, 53–106 (1995; Zbl 0838.06012)] (a strictly two-sided commutative quantale in the terminology of [K. I. Rosenthal, Quantales and their applications. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons, Inc. (1990; Zbl 0703.06007)]). More precisely, given an $$L$$-valued poset $$(X,P,E)$$, where $$X$$ is a set, and $$P$$ (resp. $$E$$) is an $$L$$-valued partial order (resp. equality) on $$X$$, the author constructs a category X as above in which the condition $$x_1\leqslant x_2$$ is replaced with $$\bot_L<P(x_1,x_2)$$. In such a way we arrive at a category $$L$$-Pos, which is claimed to be essentially the category $$\mathbf{Pos}(L)$$ (Remark 3.3 on page 152). It is easy to see, however, that such a representation is somewhat deficient, in the sense that one looses the $$L$$-valued equality $$E$$, which is a part of the objects of the category $$\mathbf{Pos}(L)$$. A better categorical framework seems to be that of double categories of [C. Ehresmann, Catégories et structures. Département de mathématiques pures et appliquées. Coll. Travaux et recherches mathématiques. 10. Paris: Dunod (1965; Zbl 0192.09803)] (see also [R. Dawson and R. Paré, Cah. Topologie Géom. Différ. Catég. 34, No. 1, 57–79 (1993; Zbl 0778.18005)]), which additionally allows the second type of morphisms to represent the missing $$L$$-valued equality $$E$$.
The author then proceeds with considering some special objects (initial, final, zero) and morphisms (epimorphisms, monomorphisms, bimorphisms and isomorphisms) in the category $$\mathbf{Pos}(L)$$ as well as (co)products of its objects. Moreover, towards the end of the paper, we find three particular lattice-valued categories induced by $$\mathbf{Pos}(L)$$ whose morphisms have a degree (an element of the $$cl$$-monoid $$L$$) attached to them, representing the extent to which their underlying maps are order-reflecting, order-preserving and order-isomorphisms, respectively. The author studies the above-mentioned categorical properties of the last two of these lattice-valued categories, and shows that the construction of generalized products of their objects has a convenient application in the theory of aggregation functions [M. Grabisch, J.-L. Marichal, R. Mesiar and E. Pap, Aggregation functions. Cambridge: Cambridge University Press (2009; Zbl 1196.00002)].
The paper is written a bit negligently (e.g., Example 4.16 on page 162 makes a referenca to a non-existing Proposition 3.15 in its proof, whereas the only sentence of Definition 4.13 on page 161 seems to lack one of its parts). Moreover, its English is slightly cumbersome. On the other hand, the paper is self-contained, easy to read, and offers a promising beginning in the development of the theory of lattice-valued categories of lattice-valued sets.
##### MSC:
 03E72 Theory of fuzzy sets, etc. 06F07 Quantales 18A05 Definitions and generalizations in theory of categories 18B35 Preorders, orders, domains and lattices (viewed as categories)
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