On linearly ordered <Emphasis Type="Italic">H</Emphasis>-closed topological semilattices

We give a criterion for a linearly ordered topological semilattice to be H-closed. We also prove that any linearly ordered H-closed topological semilattice is absolutely H-closed and we show that every linearly ordered semilattice is a dense subsemilattice of an H-closed topological semilattice.     

Numbered distributive semilattices

In this article, we consider several definitions of a Lachlan semilattice; i.e., a semilattice isomorphic to a principal ideal of the semilattice of computably enumerable m-degrees. We also answer a series of questions on constructive posets and prove that each distributive semilattice with top and bottom is a Lachlan semilattice if it admits a Σ ₃ ⁰ -representation as an algebra but need not be a Lachlan semilattice if it admits a Σ ₃ ⁰ -representation as a poset. The examples are constructed of distributive lattices that are constructivizable as posets but not constructivizable as join (meet) semilattices. We also prove that every locally lattice poset (in particular, every lattice and every distributive semilattice) possessing a Δ ₂ ⁰ -representation is positive.     

On the Least Property of the Semilattice Congruences on PO-Semigroups

n on po-semigroups. We study the least property of (ordered) semilattice congruences, and prove: 1. N is the least ordered semilattice congruence on pr-semigroups (cf.[1]). 2. n is the least semilattice congruence on po-semigroups. 3. N is not the least semilattice congruence on po-semigroups in general. Thus, we give a complete solution to the problem posed by N. Kehayopulu in [1].     

Semigroups of right quotients of a semigroup which is a semilattice of groups

Let S be a semigroup with zero which is a semilattice of groups. In [6], McMorris showed that the semigroup of quotients Q=Q(S) corresponding to the filter of “dense” right ideals of the semigroup S is also a semilattice of groups. He accomplished this by noting that Q is a regular semigroup in which all idempotents are central, an equivalent formulation of a semilattice of groups. In this paper we develop the semigroup of quotients Q corresponding to an arbitrary right quotient filter on S (as defined herein) and note the above result in this more general setting by explicitly constructing a semigroup which is isomorphic to Q. We also see that the underlying semilattice for Q in this case is isomorphic to a semigroup of quotients of the original semilattice for the semigroup S.     

Hilbert algebras as implicative partial semilattices

The infimum of elements a and b of a Hilbert algebra are said to be the compatible meet of a and b, if the elements a and b are compatible in a certain strict sense. The subject of the paper will be Hilbert algebras equipped with the compatible meet operation, which normally is partial. A partial lower semilattice is shown to be a reduct of such an expanded Hilbert algebra i ?both algebras have the same ?lters.An expanded Hilbert algebra is actually an implicative partial semilattice (i.e., a relative subalgebra of an implicative semilattice),and conversely.The implication in an implicative partial semilattice is characterised in terms of ?lters of the underlying partial semilattice.     

Semilattice modes I: the associated semiring

We examine idempotent, entropic algebras (modes) which have a semilattice term. We are able to show that any variety of semilattice modes has the congruence extension property and is residually small. We refine the proof of residual smallness by showing that any variety of semilattice modes of finite type is residually countable. To each variety of semilattice modes we associate a commutative semiring satisfying 1 +r=1 whose structure determines many of the properties of the variety. This semiring is used to describe subdirectly irreducible members, clones, subvariety lattices, and free spectra of varieties of semilattice modes.Presented by J. Berman.Part of this paper was written while the author was supported by a fellowship from the Alexander von Humboldt Stiftung.     

Free dimonoids

We characterize the least semilattice congruence of a free dimonoid and prove that a free dimonoid is a semilattice of s-simple subdimonoids each of which is a rectangular band of subdimonoids.     

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.     

On left regular and intra-regular ordered semigroups

We study the decomposition of left regular ordered semigroups into left regular components and the decomposition of intra-regular ordered semigroups into simple or intra-regular components, adding some additional information to the results considered in [KEHAYOPULU, N.: On left regular ordered semigroups, Math. Japon. 35 (1990), 1057–1060] and [KEHAYOPULU, N.: On intra-regular ordered semigroups, Semigroup Forum 46 (1993), 271–278]. We prove that an ordered semigroup S is left regular if and only if it is a semilattice (or a complete semilattice) of left regular semigroups, equivalently, it is a union of left regular subsemigroups of S. Moreover, S is left regular if and only if it is a union of pairwise disjoint left regular subsemigroups of S. The right analog also holds. The same result is true if we replace the words “left regular” by “intraregular”. Moreover, an ordered semigroup is intra-regular if and only if it is a semilattice (or a complete semilattice) of simple semigroups. On the other hand, if an ordered semigroup is a semilattice (or a complete semilattice) of left simple semigroups, then it is left regular, but the converse statement does not hold in general. Illustrative examples are given.     

幺半群的强半格上的Rees矩阵半群的平移壳

设含幺元的半群A是幺半群A~_e的半格,其中A的幺元为1_A,A~_e的幺元为e,所有幺元e的集合为E(A),则对于幺半群A上的Rees矩阵半群S和幺半群A~_e上的Rees矩阵半群S~_e,以下五个条件是等价的:(1)任意的e∈E(A),a∈A,有ae=ea;(2)A是幺半群A~_e的强半格;(3)S是S~_e的强半格;(4)A的平移壳和A~_e的平移壳的强半格同构;(5)S的平移壳和S~_e的平移壳的强半格同构.     

On Intra-regular Ordered Semigroups

poe -semigroup -that is an ordered semigroup (:po -semigroup) having a greatest element - is a semilattice of simple semigroups if and only if it is a semilattice of simple poe -semigroups [3].     

Distributive lattices of numberings

We study into a semilattice of numberings generated by a given fixed numbering via operations of completion and taking least upper bounds. It is proved that, except for the trivial cases, this semilattice is an infinite distributive lattice every principal ideal in which is finite. The least upper and the greatest lower bounds in the semilattice are invariant under extensions in the semilattice of all numberings. Isomorphism types for the semilattices in question are in one-to-one correspondence with pairs of cardinals the first component of which is equal to the cardinality of a set of non-special elements, and the second — to the cardinality of a set of special elements, of the initial numbering. Supported by INTAS grant No. 00-429. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 83–102, January–February, 2007.     

Semilattice orders on the homomorphic images of the Rédei semigroup

The combinatorial simple principal ideal semigroups generated by two elements were described by L. Megyesi and G. Pollák. The ‘most general’ among them is called the Rédei semigroup. The ‘most special’ combinatorial simple principal ideal semigroup generated by two elements is the bicyclic semigroup. D. B. McAlister determined the compatible semilattice orders on the bicyclic semigroup. Our aim is to study the compatible semilattice orders on the homomorphic images of the Rédei semigroup. We prove that there are four compatible total orders on these semigroups. We show that on the Rédei semigroup, the total orders are the only compatible semilattice orders. Moreover, on each proper homomorphic image of the Rédei semigroup, we give a compatible semilattice order which is not a total order. Communicated by Mária B. Szendrei     

交换半群的强半格

本文刻划交换半群的强半格上的最小半格同余，并证明由此得到的商半群为对应的每个交换半群的商半群的强半格。     

Semi-lattices of bicompact <Emphasis Type="Italic">G</Emphasis>-extensions

It is shown that any semilattice of bicompact G-extensions can be realized as a semilattice for a pseudocompact phase space with an action of a discrete group. The following examples are given: lattices of bicompact G-extensions which cannot be lattices of bicompact extensions of any Tikhonov space and a pseudocompact G-space with a discrete acting group whose semilattice of bicompact G-extensions has a countable number of minimal elements.     

On irreducible elements of semilattices

The concept of (join)-irreducible elements works well, especially for distributive lattices. Therefore our definition of elements of a given degree of irreducibility employs the notion of distributivity as much as possible, even if the irreducibility is defined for elements of a (meet)-semilattice. Via the lattice of hereditary subsets of the poset ofk-irreducible elements of a semilattice (wherek is a cardinal) we obtain a new construction of a D1_k-reflection (a sort of free distributive extension) of the semilattice, provided that there are sufficiently manyk-irreducible elements. The last property is satisfied, for example, if the original semilattice is the dual of an algebraic lattice [Dilworth and Crawley, 1960], but this condition is too restrictive for semilattices. It turns out that, under certain limitations, the D1_k-reflection of a semilattice both preserves and reflects the degree of irreducibility.Presented by R. Freese.     

Dimonoids

It is proved that a system of axioms for a dimonoid is independent and Cayley’s theorem for semigroups has an analog in the class of dimonoids. The least separative congruence is constructed on an arbitrary dimonoid endowed with a commutative operation. It is shown that an appropriate quotient dimonoid is a commutative separative semigroup. The least separative congruence on a free commutative dimonoid is characterized. It is stated that each dimonoid with a commutative operation is a semilattice of Archimedean subdimonoids, each dimonoid with a commutative periodic semigroup is a semilattice of unipotent subdimonoids, and each dimonoid with a commutative operation is a semilattice of a-connected subdimonoids. Various dimonoid constructions are presented.     

The universal Lachlan semilattice without the greatest element

We deal with some upper semilattices of m-degrees and of numberings of finite families. It is proved that the semilattice of all c.e. m-degrees, from which the greatest element is removed, is isomorphic to the semilattice of simple m-degrees, the semilattice of hypersimple m-degrees, and the semilattice of Σ ₂ ⁰ -computable numberings of a finite family of Σ ₂ ⁰ -sets, which contains more than one element and does not contain elements that are comparable w.r.t. inclusion. Supported by the Grant Council (under RF President) for Young Russian Scientists via project MK-1820.2005.1. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 299–345, May–June, 2007.