Adjuncts are characteristically optional, but many, such as adverbs and adjectives, are strictly ordered. In Minimalist Grammars (MGs), it is straightforward to account for optionality or ordering, but not both. I present an extension of MGs, MGs with Adjunction, which accounts for optionality and ordering simply by keeping track of two pieces of information at once: the original category of th...

The Structural Induction Theorem (Lehmann and Smyth, 1981; Plotkin, 1981) characterizes initial F-algebras of locally continuous functors F on the category of cpo's with strict and continuous maps. Here a dual of that theorem is presented, giving a number of equivalent characterizations of nal coalgebras of such functors. In particular, nal coalgebras are order strongly-extensional (sometimes c...

The unit interval in a partially ordered abelian group with order unit forms an interval effect algebra (IEA) and can be regarded as an algebraic model for the semantics of a formal deductive logic. There is a categorical equivalence between the category of IEA’s and the category of unigroups. In this article, we study the IEA-unigroup connection, focusing on the cases in which the IEA is a Boo...

Data types may be considered as objects in any suitable category, and need not necessarily be ordered structures or many-sorted algebras. Arrays may be specified having as parameter any object from a category J.f with finite products and eoproducts, if products distribute over coproducts. The Lehmann-Smith least fixpoint approach to recursively-defined data types is extended by introducing the ...

The notion of absorbent ordered filters in implicative semigroupsis introduced, and its fuzzification is considered. Relations among (fuzzy) orderedfilters, (fuzzy) absorbent ordered filters, and (fuzzy) positive implicativeordered filters are stated. The extensionproperty for (fuzzy) absorbent orderedfilters is established. Conditions for (fuzzy) ordered filters to be (fuzzy)absorbent ordered ...

The category is introduced as an ordered 5-tuple of the form 〈O,M,dom,cod, ·, id〉 where O (objects) and M (morphisms) are arbitrary nonempty sets, dom and cod map M onto O and assign to a morphism domain and codomain, · is a partial binary map from M×M to M (composition of morphisms), id applied to an object yields the identity morphism. We define the basic notions of the category theory such a...

This paper presents a unified account of a number of dual category equivalences of relevance to the theory of canonical extensions of distributive lattices. Each of the categories involved is generated by an object having a 2-element underlying set; additional structure may be algebraic (lattice or complete lattice operations) or relational (order) and, in either case, topology may or may not b...

1.1. Sites. We begin with the underlying idea for a Grothendieck topology. Consider your favorite topological space X. Its topology, don’t use the Zariski topology, is determined by the partially ordered category UX consisting of all the open subsets of X ordered with respect to inclusion. In order to verify that a presheaf is a sheaf on X, we must first construct all possible coverings U = (Ui...

We de ne families of posets ordered by pre xes as the coun terpart of the usual families of con gurations ordered by subsets On these objects we de ne two types of morphism event and order morphisms re sulting in categories FPos and FPos We then show the following Families of posets in contrast to families of con gurations are always prime algebraic in fact the category FPos is equivalent to th...

Consider decision problems derived from optimization problems. For problems in this category, we show that they cannot be expressed in existential second order (ESO) logic with a first order part that is universal Horn (or simply, ESO Π1 Horn) at the structure level, as opposed to machine level. This is true even when we consider ordered structures. We show that an NP-complete problem (Maximum ...