On λ-semidirect products by locally ℛ-unipotent semigroups (original) (raw)

On power varieties of semigroups

Journal of Algebra, 1989

A power variety of semigroups is a variety BY generated by the semigroups of the parts of each semigroup from some variety Y. The problem of determination of the power variety z?Y associated with Y is solved in the following way. First, find the variety Stab Y defined by the "stable" identities which hold in Y. To obtain BY, take the join of Stab ,I/ with the variety of all semilattices. Stab W" is actually the variety B'Y generated by the semigroups of the nonempty parts of each member of Y. It follows that the operator P' is idempotent while B # @ = 9'. Further, Y27r is defined by the permutation identities which hold in Y. Via the manipulation of stable identities and using some basic results from the theory of well-quasi-ordering, it is shown that the semilatticc of varieties of the form BY is countable and better-quasi-ordered under inclusion. Similar questions are also considered in the context of generalized varieties.

Associativity of the regular semidirect product of existence varieties

Journal of the Australian Mathematical Society, 2000

The associativity of the regular semidirect product of existence varieties introduced by Jones and Trotter was proved under certain condition by Reilly and Zhang. Here we estabilsh associativity in many new cases. Moreover, we prove that the regular semidirect product is right distributive with respect to the join operation. In particular, both associativity and right distriutivity yiel within the varieties of completely simple semigroups. Analogous results are obtainedj for e-pseudovarieties of finite regular semigroups.

Proper restriction semigroups – semidirect products and W-products

Acta Mathematica Hungarica, 2013

Fountain and Gomes [4] have shown that any proper left ample semigroup embeds into a so-called W-product, which is a subsemigroup of a reverse semidirect product T Y of a semilattice Y by a monoid T , where the action of T on Y is injective with images of the action being order ideals of Y. Proper left ample semigroups are proper left restriction, the latter forming a much wider class. The aim of this paper is to give necessary and sufficient conditions on a proper left restriction semigroup such that it embeds into a W-product. We also examine the complex relationship between W-products and semidirect products of the form Y T .

Spined products of some semigroups

Proceedings of The Japan Academy Series A-mathematical Sciences, 1993

Communicated by Shokichi IYANAGA, M. J. A., Nov. 12, 1993) Spined products of semigroups were first defined and studied by N. Kimura, 1958, [7]. After that, spined products have been considered many a time, predominantly those of a band and a semilattice of semigroups with respect to their common semilattice homomorphic image. Spined and subdirect products of a band and a semilattice of groups are studied by M. Yamada [13],[14], J. M. Howie and G. Lallement [6] and by M. Petrich [10]; spined products of a band and some types of semilattices of monoids are studied by F. Pastijn [8], A. El-Qallali [3],[4], and by R. J. Warne [12]. For other considerations of these products we refer to [4],[5],[7],[9], . In the quoted papers, spined products are considered in connection with some types of bands of semigroups. In this paper we give a general composition for bands of semigroups that are (punched) spined products of a band and a semilattice of semigroups. This composition, in some sence, is a generalization of a well-known semilattice composition (see Theorem III 7.2. [9]).

Equations on the semidirect product of a finite semilattice by a finite commutative monoid

Semigroup Forum, 1994

Let Com t , q denote the variety of finite monoids that satisfy the equations xy = yx and x t = x t+q. The variety Com 1,1 is the variety of finite semilattices also denoted by J 1. In this paper, we consider the product variety J 1 *Com t,q generated by all semidirect products of the form M * N with M J 1 and N Com t,q. We give a complete sequence of equations for J 1 * Com t,q implying complete sequences of equations for J 1 * (Com A), J 1 * (Com G) and J 1 * Com, where Com denotes the variety of finite commutative monoids, A the variety of finite aperiodic monoids and G the variety of finite groups. Article: 1. Introduction Let Com t , q denote the variety of finite monoids that satisfy the equations xy = yx and x t = x t+q. The variety Com 1,1 is the variety of finite semilattices also denoted by J 1. In this paper, we give an equational characterization of the product variety J 1 * Com t , q generated by all semidirect products of the form M * N with M J 1 and N Com t , q. Our results imply a complete sequence of equations for J 1 * (Com A), J 1 *(Com G) and J 1 *Com, where Com denotes the variety of finite commutative monoids, A the variety of finite aperiodic monoids and G the variety of finite groups. Pin [12] has shown that the variety J 1 * Com 1 , 1 is defined by the equations xux = xux 2 and xuyvxy = xuyvyx. Irastorza [7] has given equations of the particular products J 1 * Com 0,q and has shown that, although the two varieties J 1 and Com 0 , 2 are defined by finite sequences of equations, their product is not. Almeida [1] has given an equational characterization of the variety of finite monoids generated by all semidirect products of i finite semilattices and has shown that it is defined by a finite sequence of equations if and only if i = 1 or 2. Ash [2] has shown that the variety J 1 * G = Inv is defined by the equation x w y w = y w x w , that is, J 1 * G is the variety generated by the inverse semigroups. Our results follow from versions of techniques used in particular by Blanchet-Sadri [3], Brzozowski and Simon [4] and Pin [11, 12]. 1.1. Definitions and notations Let M and N be monoids. We say that M divides N and write M N if M is a morphic image of a submonoid of N. Note that the divisibility relation is transitive. An M-variety V is a family of finite monoids that satisfies the following two conditions:  If N V and M N , then M V .

A Note on Semilattice Decompositions of Completely …Regular Semigroups

Ciric2 Abstract. We study completely …-regular semigroups admitting a de- composition into a semilattice of æn-simple semigroups, and describe them in terms of properties of their idempotents. In the general case, semi- groups admitting a decomposition into a semilattice of æn-simple semi- groups were characterized by M. ´ Ciric and S. Bogdanovic in (3) (see The- orem 1 below), in terms of paths of length n in the graph corresponding to the relation ¡!, and in terms of principal filters and n-radicals. Here we prove that in the completely …-regular case, it suces to consider only those paths of length n starting and/or ending with and idempotent, as well as principal filters and n-radicals generated by idempotents.

Biunit pairs in semiheaps and associated semigroups

Semigroup Forum

The notion of neutral element generalizes to a pair of elements in ternary algebras. Biunit pairs are introduced as pairs of elements in a semiheap that generalize the notion of Mal’cev element. In order to generalize the known correspondences between semiheaps and certain kinds of semigroups, families of functions generalizing involutions and conjugations, called switches and warps, are investigated. The main theorem establishes that there is a one-to-one correspondence between monoids equipped with a particular switch and semiheaps with a fixed biunit pair. This generalizes the celebrated result in semiheap theory that gives a one-to-one correspondence between involuted monoids and semiheaps with a fixed biunit element. A novel, previously undocumented, algebra is motivated by this result: diheaps are introduced as semiheaps whose elements belong to biunit pairs, which generalize the well-known case of heaps. Diheaps are of great interest since they are shown to be isomorphic to h...

Some algebraic structures on the generalization general products of monoids and semigroups

Arabian Journal of Mathematics, 2020

For arbitrary monoids A and B, in Cevik et al. (Hacet J Math Stat 2019:1–11, 2019), it has been recently defined an extended version of the general product under the name of a higher version of Zappa products for monoids (or generalized general product) A^{\oplus B}A⊕BA ⊕ BA⊕B_{\delta }\bowtie _{\psi }B^{\oplus A}$$ δ ⋈ ψ B ⊕ A and has been introduced an implicit presentation as well as some theories in terms of finite and infinite cases for this product. The goals of this paper are to present some algebraic structures such as regularity, inverse property, Green’s relations over this new generalization, and to investigate some other properties and the product obtained by a left restriction semigroup and a semilattice.