A Note on C-wrpp Semigroups

ＬｅｔＲｄｅｎｏｔｅＧｒｅｅｎ ｓＲ ｒｅｌａｔｉｏｎｏｎａｓｅｍｉｇｒｏｕｐＳ .ＷｅｄｅｆｉｎｅａｒｉｇｈｔｃｏｎｇｒｕｅｎｃｅＬ ｏｎＳｉｎｔｈｅｒｕｌｅ:ｆｏｒａ ,ｂ∈Ｓ ,(ａ ,ｂ) ∈Ｌ ｉｆｆｆｏｒａｌｌｘ ,ｙ∈Ｓ1,　　 (ａｘ ,ａｙ) ∈Ｒ (ｂｘ,ｂｙ) ∈Ｒ .ＡｓｅｍｉｇｒｏｕｐＳｉｓｓａｉｄｔｏｂｅａＣ ｗｒｐｐｓｅｍｉｇｒｏｕｐｉｆａｌｌｉｄｅｍｐｏｔｅｎｔｓｏｆＳａｒｅｃｅｎｔｒａｌｉｎＳａｎｄｅａｃｈＬ ｃｌａｓｓｏｆＳｃｏｎｔａｉｎｓａｎｉｄｅｍｐｏｔｅｎｔ(ｓｅｅ [1 ] ) .　Ｔｈｅｐ…     

Maximal and Minimal Semilattices on Ordered Sets

The map which takes an element of an ordered set to its principal ideal is a natural embedding of that ordered set into its powerset, a semilattice. If attention is restricted to all finite intersections of the principal ideals of the original ordered set, then an embedding into a much smaller semilattice is obtained. In this paper the question is answered of when this construction is, in a certain arrow-theoretic sense, minimal. Specifically, a characterisation is given, in terms of ideals and filters, of those ordered sets which admit a so-called minimal embedding into a semilattice. Similarly, a candidate maximal semilattice on an ordered set can be constructed from the principal filters of its elements. A characterisation of those ordered sets that extend to a maximal semilattice is given. Finally, the notion of a free semilattice on an ordered set is given, and it is shown that the candidate maximal semilattice in the embedding-theoretic sense is the free object.     

Algebraic coalitions: 2

This paper continues the investigation of [2] into the notions of algebra in and of coalition. Here these notions are characterized for the variety of semilattices, more generally for any variety of inverse semigroups, and for subtractive varieties. Received June 25, 1997 accepted in final form October 16, 1998.     

Some remarks on completion of numberings

We present some examples and constructions in the theory of complete numberings and completions of numberings that partially answer a few questions of [1].     

Algebras Defined from Ordered Sets and the Varieties they Generate

We investigate ways of representing ordered sets as algebras and how the order relation is reflected in the algebraic properties of the variety (equational class) generated by these algebras. In particular we consider two different but related methods for constructing an algebra with one binary operation from an arbitrary ordered set with a top element. The two varieties generated by all these algebras are shown to be well-behaved in that they are locally finite, finitely based, and have an equationally definable order relation. We exhibit a bijection between the subdirectly irreducible algebras in each variety and the class of all ordered sets with top element. We determine the structure and cardinality of the free algebra on n-free generators and provide sharp bounds on the number of n-generated algebras in each variety. These enumeration results involve the number of quasi-orders on an n-element set.     

关于序半群的N—类

本文给出了关于序半群 N—类的若干等价性质     

Degrees of presentability of structures. I

Presentations of structures in admissible sets, as well as different relations of effective reducibility between the structures, are treated. Semilattices of degrees of Σ-definability are the main object of investigation. It is shown that the semilattice of degrees of Σ-definability of countable structures agrees well with semilattices of T-and e-degrees of subsets of natural numbers. Also an attempt is made to study properties of the structures that are inherited under various effective reducibilities and explore how degrees of presentability depend on choices of different admissible sets as domains for presentations. Supported by RFBR grant Nos. 05-0100481 and 06-0104002, by the Council for Grants (under RF President) for State Support of Young Candidates of Science and Their Supervisors via project MK-1239.2005.1, and via INTAS project YSF 04-83-3310. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 763–788, November–December, 2007.     

逆半群半格上的自然偏序关系

首先讨论逆半群Sa的半上的自然偏序与各Sa上的自然偏序之间的关系，然后讨论逆半群Sa的半格S的幂等元集Ea的闭包限制在Sa上与Ea的闭包的关系．     

Non-Complementedness and Non-Distributivity of Kleene Degrees

In this note, we study the complementedness and the distributivity of upper semilattices of Kleene degrees assuming V = L. K denotes the upper semilattice of all Kleene degrees. We prove that if V = L, then some sub upper semilattices of K are non-complemented and some are non-distributive.