Varieties Of Semigroups Research Articles

Identities and quasiidentities in the lattice of overcommutative semigroup varieties

We describe overcommutative varieties of semigroups whose lattice of overcommutative subvarieties satisfies a non-trivial identity or quasiidentity. These two properties turn out to be equivalent.

Modular and lower-modular elements of lattices of semigroup varieties

The paper contains three main results. First, we show that if a commutative semigroup variety is a modular element of the lattice Com of all commutative semigroup varieties then it is either the variety \(\mathcal{COM}\) of all commutative semigroups or a nilvariety or the join of a nilvariety with the variety of semilattices. Second, we prove that if a commutative nilvariety is a modular element of Com then it may be given within \(\mathcal{COM}\) by 0-reduced and substitutive identities only. Third, we completely classify all lower-modular elements of Com. As a corollary, we prove that an element of Com is modular whenever it is lower-modular. All these results are precise analogues of results concerning modular and lower-modular elements of the lattice of all semigroup varieties obtained earlier by Ježek, McKenzie, Vernikov, and the author. As an application of a technique developed in this paper, we provide new proofs of the ‘prototypes’ of the first and the third our results.

Proofs of definability of some varieties and sets of varieties of semigroups

We show that many important varieties and sets of varieties of semigroups may be defined by relatively simple and transparent first-order formulas in the lattice of all semigroup varieties.

Distributive hyperidentities in semigroups

The paper gives a description of the class of semigroups in which the hyperidentity of left distributivity is essentially valid. It is proved that the class of all semigroups, in which this hypeidentity is essentially valid, is a union of three finitely based semigroup varieties, and the basic identities of all varieties are given in explicit forms.

A common framework for restriction semigroups and regular ∗ -semigroups

Left restriction semigroups have appeared at the convergence of several flows of research, including the theories of abstract semigroups, of partial mappings, of closure operations and even in logic. For instance, they model unary semigroups of partial mappings on a set, where the unary operation takes a map to the identity map on its domain. This perspective leads naturally to dual and two-sided versions of the restriction property. From a varietal perspective, these classes of semigroups–more generally, the corresponding classes of Ehresmann semigroups–derive from reducts of inverse semigroups, now taking a to a + = a a − 1 (or, dually, to a ∗ = a − 1 a , or in the two-sided version, to both). In this paper the notion of restriction semigroup is generalized to P -restriction semigroup, derived instead from reducts of regular ∗-semigroups (semigroups with a regular involution). Similarly, [left, right] Ehresmann semigroups are generalized to [left, right] P -Ehresmann semigroups. The first main theorem is an abstract characterization of the posets P of projections of each type of such semigroup as ‘projection algebras’. The second main theorem, at least in the two-sided case, is that for every P -restriction semigroup S there is a P -separating representation into a regular ∗-semigroup, namely the ‘Munn’ semigroup on its projection algebra, consisting of the isomorphisms between the algebra’s principal ideals under a modified composition. This theorem specializes to known results for restriction semigroups and for regular ∗-semigroups. A consequence of this representation is that projection algebras also characterize the posets of projections of regular ∗-semigroups. By further characterizing the sets of projections ‘internally’, we connect our universal algebraic approach with the classical approach of the so-called ‘York school’. The representation theorem will be used in a sequel to show how the structure of the free members in some natural varieties of ( P -)restriction semigroups may easily be deduced from the known structure of associated free inverse semigroups.

Codistributive elements of the lattice of semigroup varieties

We prove that if a semigroup variety is a codistributive element of the lattice SEM of all semigroup varieties, then it either coincides with the variety of all semigroups or is a variety of semigroups with completely regular square. We completely classify strongly permutable varieties that are codistributive elements of SEM.We prove that a semigroup variety is a costandard element of the lattice SEMif and only if it is a neutral element of this lattice. In view of results obtained earlier, this gives a complete description of costandard elements of the lattice SEM.

Distributive and neutral elements of the lattice of commutative semigroup varieties

We completely describe commutative semigroup varieties that are distributive, standard, or neutral elements of the lattice of all commutative semigroup varieties. In particular, we prove that the properties of being a distributive element and of being a standard element in this lattice are equivalent.

Associative nontrivial hyperidentities in semigroups

The paper gives a characterization of the class of semigroups in which the nontrivial associative hyperidentity is essentially satisfied. It is proved that, in difference to the case of trivial associative hyperidentity, the class of all semigroups, in which a nontrivial associative hyperidentity is essentially satisfied, is a finite union of finitely based varieties of semigroups, and the basis identities of all varieties are explicitly inscribed.

ON MAXIMAL SUBGROUPS OF FREE OBJECTS OF CERTAIN COMPLETELY REGULAR SEMIGROUP VARIETIES

By adjusting a method of Kadourek and Polák developed for free semigroups satisfying xr≏ x, we prove that if [Formula: see text] is a periodic group variety, then any maximal subgroup of the free object in the completely regular semigroup variety of the form [Formula: see text] is a relatively free group in [Formula: see text] over a suitable set of free generators. When [Formula: see text] is locally finite, we provide some bounds for the sizes of its finitely generated members.

An equimorphic diversity case

B.M. Schein has proved that in the semigroup variety NB = Mod ( x 2 = x , x u v y = x v u y ) of normal bands one can find at most four pairwise non-isomorphic semigroups with isomorphic monoids of endomorphisms. We prove here the same result for the extension semigroup variety NB ˜ = Mod ( x 2 y = x y , x u v y = x v u y ) of NB , properly separated from NB only by the variety of inflations of the NB -semigroups.

GENERALIZED CONTEXTS AND n-ARY SYNTACTIC SEMIGROUPS OF TREE LANGUAGES

A new type of syntactic monoid and semigroup of tree languages is introduced. For each n ≥ 1, we define for any tree language T its n-ary syntactic monoid Mn(T) and its n-ary syntactic semigroup Sn(T) as quotients of the monoid or semigroup, respectively, formed by certain new generalized contexts. For n = 1 these contexts are just the ordinary contexts (or 'special trees') and M1(T) is the syntactic monoid introduced by W. Thomas (1982,1984). Several properties of these monoids and semigroups are proved. For example, it is shown that Mn(T) and Sn(T) are isomorphic to certain monoids and semigroups associated with the minimal tree recognizer of T. Using these syntactic monoids or semigroups, we can associate with any variety of finite monoids or semigroups, respectively, a variety of tree languages. Although there are varieties of tree languages that cannot be obtained this way, we prove that the definite tree languages can be characterized by the syntactic semigroups S2(T), which is not possible using the classical syntactic monoids or semigroups.

LIMIT VARIETIES GENERATED BY COMPLETELY 0-SIMPLE SEMIGROUPS

A limit variety is a variety that is minimal with respect to being nonfinitely based. This paper presents a new infinite series of limit semigroup varieties, each of which is generated by a finite 0-simple semigroup with Abelian subgroups. These varieties exhaust all limit varieties generated by completely 0-simple semigroups with Abelian subgroups.

Identities determining varieties of semigroups with completely regular power

We provide a transparent syntactic algorithm to decide whether an identity defines a variety of semigroups with completely regular power.

Pre-strongly solid varieties of commutative semigroups

Pre-strongly solid varieties of commutative semigroups

On the algorithmic decidability of the second-type annihilation predicate for varieties of semigroups

The object of this study is the second-type annihilation predicate α: (a, b) ∈ a ⇔ ab = ba = a, b2 = b. In terms of identities, we give a complete description of varieties of semigroups which are finitely approximable with respect to this predicate. Another subject considered in this work is the problem of algorithmic decidability of the predicate. Solution of this problem is related to finite semigroups the absence of which in the variety is the condition for existence of the corresponding algorithm. Our proof relies on results of M. Petrich, A.I. Mal’tsev, E.A. Golubov, M.V. Sapir, and S.I. Kublanovskii.

Erratum to: “Special Elements of the Lattice of Overcommutative Varieties of Semigroups”

Erratum to: “Special Elements of the Lattice of Overcommutative Varieties of Semigroups”

Lower-modular elements of the lattice of semigroup varieties. III

Lower-modular elements of the lattice of semigroup varieties. III

Research Concerning the Behaviour of Some Potato Varieties in Different Photoperiodic Conditions

The length of the day, in the geographical area conditions of Romania, is different from other geographical areas where the potato is cultivated in the same calendar periods. In the specialized literature, there are frequent references to short days influence, between 12 and 14 hours, on potato tuberization, with influences more and more favourable. Considering the calculations for Romania’s conditions, the length of the day has values less than 14 hours only in March, period that does not coincide with the period of potato tuberization. The period in which the potato tubers in Romania is generally between 15 and 25 May, when the length of the day passes 15 hours. Generally, in the conditions of a normal spring plantation, between 1 and 20 April, tuberization take place in the conditions of end May, when the length of the day passes 15 hours (21. 05.200715 h 15min, 30.05.2008 – 15h 34 min). In our conducted studies we have tested four varieties of potato, two from two different groups of maturity (Ostara and group varieties Impala and semi early Laura Redsec and group varieties semi late). Research has been conducted under the earliness aspect at Rascruci, Cluj County, in a greenhouse and under late planting aspects in the field of experimental USAMV from ClujNapoca Jucu commune, Cluj County. For a tuberization in terms of long days of the year the experiments have been planted in high altitude in the experimental field of USAMV Cluj located at a glacier, Gârda village, Alba County (in the Apuseni Mountains, the Plateau Calineasa the perimeter of the village Gheţar) Locating the experiments in three different areas have targeted first of all the influence of the photoperiod on tuberization process and its interactions with the soil and advancing the vegetation through the germination of the tubers for seed and early planting in the plastic greenhouse. Experiments location was done using the method of subdividing plots with three factors in three repetitions. Tuberization determinations through number and mass of the tubers formed on fractions of 5-10 mm, 10-20 mm, 20-30 mm and over 30 mm were quantified at 25, 35 and 45 days from the emergence, in the aim of better appreciations of the experimental factors and their interactions with the environmental conditions. As an average of the three experimental years, it could be said that superior results are obtained in the plastic greenhouse regarding the number of tubers/plant comparative to field results, certifying the fact that the potato plant (average results in semi-early and semi-late cultivars) is responding favourably to a lower illumination period.

Distributive elements of the lattice of semigroup varieties

A complete description is given for semigroup varieties, which are distributive elements of the lattice of all semigroup varieties.

Special Elements in the Lattice of Overcommutative Semigroup Varieties Revisited

We completely determine all distributive, codistributive, standard, costandard, and neutral elements in the lattice of overcommutative semigroup varieties, thus correcting a gap in a previous paper.