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Ultrafilter extensions of arbitrary first-order models were introduced in Saveliev (2012). The main precursor of this construction was the extension of semigroups to semigroups of ultrafilters, a technique that was used to obtain significant results in algebra and dynamics. Here we consider another particular case where the models are linearly ordered sets. We explicitly calculate the extensions of...
A well-known result by P. Cameron provides us with a construction of the free group of rank 2 ℵ 0 within the automorphism group of the rationals. We show that the full versatility of doubly transitive automorphism groups is not necessary by extending Cameron’s construction to a larger class of permutation groups and we generalize his result by constructing ...
In his monograph, H. Gonshor showed that Conway’s real closed field of surreal numbers carries an exponential and logarithmic map. In this paper, we give a complete description of the exponential equivalence classes in the spirit of the additive and multiplicative equivalence classes. This description is made in terms of a recursive formula as well as a sign sequence formula for the family of representatives...
Felsner and Reuter introduced the linear extension diameter of a partially ordered set P , denoted led( P ), as the maximum distance between two linear extensions of P , where distance is defined to be the number of incomparable pairs appearing in opposite orders (reversed) in the linear extensions. In this paper, we introduce the reversal ratio RR...
The goal of this paper is to generalize some of the existing toolkit of combinatorial algebraic topology in order to study the homology of abstract chain complexes. We define shellability of chain complexes in a similar way as for cell complexes and introduce the notion of regular chain complexes. In the case of chain complexes coming from simplicial complexes we recover...
We introduce the notion of F-augmented closure spaces by incorporating an additional structure (a family of finite subsets of the underlying set) into a given closure space in an appropriate way. We also introduce the notion of F-morphisms between F-augmented closure spaces and establish the equivalence between the category of F-augmented closure spaces and that of algebraic domains with Scott continuous...
Combinatorial structures have been considered under various orders, including substructure order and homomorphism order. In this paper, we investigate the homomorphic image order, corresponding to the existence of a surjective homomorphism between two structures. We distinguish between strong and induced forms of the order and explore how they behave in the context of different common combinatorial...
A reconstruction problem is formulated for Sperner systems, and infinite families of non-reconstructible Sperner systems are presented. This has an application to a reconstruction problem for functions of several arguments and identification minors. Sperner systems being representations of certain monotone functions, infinite families of non-reconstructible functions are thus obtained. The clones...
In this article we show the equivalence of QRB, QFS, and compact quasicontinuous domains. QRB and QFS domains are generalizations of RB and FS domains to the setting of quasicontinuous domains and compactness means compactness in the Lawson topology. This equivalence extends in the algebraic setting to a quasicontinuous version of bifinite domains. The Smyth powerdomain is a basic tool in the proofs,...
This note gives a complete characterization of when the ordinal sum of two lattices (the lattice obtained by placing the second lattice on top of the first) is projective. This characterization applies not only to the class of all lattices, but to any variety of lattices, and in particular, to the class of distributive lattices. Lattices L with the property that every epimorphism onto L has an isotone...
We show that there are uncountably many countable homogeneous lattices. We give a discussion of which such lattices can be modular or distributive. The method applies to show that certain other classes of structures also have uncountably many homogeneous members.
As is well-known, every generalized effect algebra can be embedded as a maximal proper ideal in an effect algebra called its unitization. We show that a necessary and sufficient condition that a generalized pseudo effect algebra can similarly be embedded as a maximal proper ideal in a pseudo effect algebra is that it admits a so-called unitizing automorphism. On the other hand, we show that a pseudo...
We consider the number of linear extensions of an N-free order P. We give upper and lower bounds on this number in terms of parameters of the corresponding arc diagram. We propose a dynamic programming algorithm to calculate the number. The algorithm is polynomial if a new parameter called activity is bounded by a constant. The activity can be bounded in terms of parameters of the arc diagram.
Euclidean functions with values in an arbitrary well-ordered set were first considered by Motzkin in 1949 and studied in more detail by Samuel and Nagata in the 1970’s and 1980’s. Here these results are revisited, simplified, and extended. The main themes are (i) consideration of Ord-valued functions on an Artinian poset and (ii) use of ordinal arithmetic, including the Hessenberg-Brookfield ordinal...
The cover-incomparability graph of a poset P is the edge-union of the covering and the incomparability graph of P. In this paper, the cographs that are cover-incomparability graphs are characterized, and the forbidden isometric subposet characterization of the posets whose cover-incomparability graphs are cographs is proved. In addition, some properties of the cover-incomparability chordal graphs...
The Kalmbach monad is the monad that arises from the free-forgetful adjunction between bounded posets and orthomodular posets. We prove that the category of effect algebras is isomorphic to the Eilenberg-Moore category for the Kalmbach monad.
In earlier work the second author introduced the tool of pointwise directed families of characteristic functions to establish that certain continuous function spaces [X → Q] were continuous domains. In this paper we extend those earlier results, while significantly simplifying and refining the machinery involved. Our main result asserts that the function space [X → Q] is a continuous dcpo if X is...
We study the relation on linear orders induced by order preserving surjections. In particular we show that its restriction to countable orders is a bqo.
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