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.     

On the structure of dimonoids

J. -L. Loday introduced the notion of a dimonoid and constructed the free dimonoid. Cayley’s theorem for dimonoids states that every dimonoid is isomorphic to some transformation dimonoid. In this paper we propose another approach to constructing dimonoids which is based on using a semigroup operation. Several dimonoid-theoretical constructions are suggested, and it is shown that any dimonoid is isomorphically embedded into some dimonoid constructed from a semigroup. A similar result is obtained for dirings.     

Structure of relatively free dimonoids

Loday introduced the notion of a dimonoid and constructed the free dimonoid. The paper concerns the variety theory of dimonoids. We recall and summarize the results obtained by Loday, the author as well as others on the structure of some relatively free dimonoids. This part of the paper should be viewed as a survey, however we include some new ideas in the last sections. Namely, we construct a free (?z;rs)-dimonoid, a free (rs;rz)-dimonoid, a free rs-dimonoid, a free (rb;rs)-dimonoid, a free (rs;rb)-dimonoid and characterize the least (?z;rs)-congruence, the least (rs;rz)-congruence, the least rs-congruence, the least (rb;rs)-congruence and the least (rs;rb)-congruence on the free dimonoid. We also establish that the automorphism groups of constructed free algebras are isomorphic to the symmetric group and the semigroups of the free rs-dimonoid are anti-isomorphic.     

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.     

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.     

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.     

Automorphisms of the endomorphism semigroup of a free commutative dimonoid

We determine all isomorphisms between the endomorphism semigroups of free commutative dimonoids and prove that all automorphisms of the endomorphism semigroup of a free commutative dimonoid are quasi-inner. In particular, we answer a question of B. I. Plotkin.     

Certain congruences on free trioids

Loday and Ronco introduced the notion of a trioid and constructed the free trioid of rank 1. This paper is devoted to the study of congruences on trioids. We characterize the least dimonoid congruences and the least semigroup congruence on the free (commutative, rectangular) trioid.     

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.     

Idempotent semirings with a commutative additive reduct

For every semigroup S , we define a congruence relation ρ on the power semiring (P(S),\cup,\circ) of S . If S is a band, then P(S)/ρ is an idempotent semiring . This enables us to find models for the free objects in the variety of idempotent semiring s whose additive reduct is a semilattice. December 28, 1999     

n-median semilattices

An n-median semilattice (n3) is a meet-semilattice such that (i) every principal ideal is a distributive lattice and (ii) any n-element set of elements is bounded above whenever each of its (n-1)-element subsets has an upper bound. A 3-median semilattice is thus a median semilattice in the classical sense. In this note we demonstrate how the characteristic features of median semilattices carry over to the more general case of n-median semilattices.Research supported in part by ONR Grant N00014-90-1008.     

The Lattice of Convex Subsemilattices of a Semilattice

Abstract. The convex subsemilattices of a semilattice E form a lattice C o(E) in the natural way. The purpose of this paper is to study how the properties of this lattice relate to the semilattice itself. For instance, lower semimodularity of the lattice is equivalent, along with various properties, to the semilattice being a tree. When E has more than two elements the lattice does, however, fail many common lattice-theoretic tests. It turns out that it is more fruitful to describe those semilattices E for which every ``atomically generated' filter of C o(E) satisfies certain lattice-theoretic properties.     

Quasivarieties of Sugihara semilattices with involution

It is shown that the lattice of quasivarieties contained in the quasivariety generated by an n-element relatively subdirectly irreducible Sugihara semilattice with involution contains a sublattice isomorphic to the ideal lattice of a free lattice iff n≥3. Two consequences of the result are mentioned. In memory of Victor A. Gorbunov Translated fromAlgebra i Logika, Vol. 39, No. 1, pp. 47–65, January–February, 2000.     

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.     

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.     

One Hundred Twenty-Seven Subsemilattices and Planarity

Let L be a finite n-element semilattice. We prove that if L has at least 127 ? 2^n??8 subsemilattices, then L is planar. For n >?8, this result is sharp since there is a non-planar semilattice with exactly 127 ? 2^n??8 ??1 subsemilattices.

     

On the free implicative semilattice extension of a Hilbert algebra

Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.     

A semilattice of numberings. II

\mathfrakc \mathfrak{c} -Universal semilattices \mathfrakA \mathfrak{A} of the power of the continuum (of an upper semilattice of m-degrees ) on admissible sets are studied. Moreover, it is shown that a semilattice of \mathbbH\mathbbF( \mathfrakM ) \mathbb{H}\mathbb{F}\left( \mathfrak{M} \right) -numberings of a finite set is \mathfrakc \mathfrak{c} -universal if \mathfrakM \mathfrak{M} is a countable model of a c-simple theory.