Our work proposes a new paradigm for the study of various classes of cancellative residuated lattices by viewing these structures as lattice-ordered groups with a suitable operator (a conucleus). One consequence of our approach is the categorical equivalence between the variety of cancellative commutative residuated lattices and the category of abelian lattice-ordered groups endowed with a conu...

If S is an order-adjoint monad, that is, a monad on Set that factors through the category of ordered sets with left adjoint maps, then any monad morphism τ : S → T makes T orderadjoint, and the Eilenberg-Moore category of T is monadic over the category of monoids in the Kleisli category of S.

The following discusses a generalization of Baire category in a continuous set. The objective is to provide a meaningful classification of subsets of a continuous set as “large” or “small” sets in linearly ordered continuous sets. In particular, for cardinal number κ, the continuous ordered set 2∗ a subset of the set of dyadic sequences of length κ is discussed. We establish that this space, an...

• For square contingency tables with ordered categories, this paper proposes some distance subsymmetry models. The one model indicates that the cumulative probability that an observation will fall in row category i or below and column category i + k (k ≥ 2) or above, is equal to the probability that it falls in column category i or below and row category i+k or above. This paper also gives the ...

Starting from the symmetric group Sn , we construct two fiat 2-categories. One of them can be viewed as the fiat “extension” of the natural 2-category associated with the symmetric inverse semigroup (considered as an ordered semigroup with respect to the natural order). This 2-category provides a fiat categorification for the integral semigroup algebra of the symmetric inverse semigroup. The ot...

Sirs—One of us (SRC) coauthored a paper in the International Journal of Epidemiology that detailed a use of data expansion to fit the continuation-ratio model1 as sketched by Armstrong and Slone,2 as well as a paper in the Annals of Epidemiology that detailed a related use of data expansion to fit proportional-odds models.3 In fact, a family of ordered logistic regression models, including cont...

Applying (enriched) categorical structures we define the notion of ordered sheaf on a quantaloid Q, which we call ‘Q-order’. This requires a theory of semicategories enriched in the quantaloid Q, that admit a suitable Cauchy completion. There is a quantaloid Idl(Q) of Q-orders and ideal relations, and a locally ordered category Ord(Q) of Q-orders and monotone maps; actually, Ord(Q) = Map(Idl(Q)...

As a practical foundation for a homotopy theory of abstract spacetime, we propose a convenient category S , which we show to extend a category of certain compact partially ordered spaces. In particular, we show that S ′ is Cartesian closed and that the forgetful functor S →T ′ to the category T ′ of compactly generated spaces creates all limits and colimits.