Nil Semigroup Research Articles

A generalized variety problem

Let $${\mathscr {C}}\! om $$ denote the variety of all commutative semigroups. For $$n\ge 1$$ let $${{\mathscr {N}}\! il }_n$$ (respectively $${\mathscr {N}}_n$$ ) denote the variety of all nil (respectively nilpotent) semigroups of index n, and let $${\mathscr {N}}\! il $$ (respectively $${\mathscr {N}}$$ ) denote the generalized variety of all nil (respectively nilpotent) semigroups. For a class $${\mathscr {W}}$$ of semigroups let $${\mathscr {L}}({\mathscr {W}})$$ denote the lattice of all varieties contained in $${\mathscr {W}}$$ , and let $${\mathscr {G}}({\mathscr {W}})$$ denote the lattice of all generalized varieties contained in $${\mathscr {W}}$$ . Almeida has proved that the map $$\alpha :{\mathscr {L}}({\mathscr {N}}\! il \cap {\mathscr {C}}\! om )\cup \{{\mathscr {N}}\! il \cap {\mathscr {C}}\! om \}\rightarrow {\mathscr {G}}({\mathscr {N}}\cap {\mathscr {C}}\! om )$$ given by $${\mathscr {W}}\alpha = {\mathscr {W}}\cap {\mathscr {N}}$$ is an isomorphism, and asked whether the extension $${\alpha }': {\mathscr {L}}({\mathscr {N}}\! il ) \cup \{{\mathscr {N}}\! il \}\rightarrow {\mathscr {G}}({\mathscr {N}})$$ of this map is also an isomorphism. The author has previously shown that $${\alpha }'$$ is not injective, using varieties of nil index 5, and noted that the corresponding map $${\alpha }_n : {\mathscr {L}}({{\mathscr {N}}\! il }_n) \rightarrow {\mathscr {G}}({\mathscr {N}} \cap {{\mathscr {N}}\! il }_n)$$ given by $${\mathscr {W}} {\alpha }_n ={\mathscr {W}}\cap {\mathscr {N}}$$ is therefore not injective for $$n\ge 5$$ . The question of smaller values of n was left open, and in this article it is shown that $${\alpha }_n$$ is not injective for $$n\ge 2$$ , using Leech’s square-free words.

Construction of Infinite Finitely Presented Nilsemigroup

This paper is devoted to construction of finitely presented infinite nil semigroup with identity $x^9=0$. This construction answers to the problem of Lev Shevrin and Mark Sapir. This is the short version of the paper presenting the main ideas of the construction. The work was carried out with the help of the Russian Science Foundation Grant N 17-11-01377. The first author is the winner of the contest Young Mathematics of Russia .

Sub-semigroups determined by the zero-divisor graph

In this paper we study sub-semigroups of a finite or an infinite zero-divisor semigroup S determined by properties of the zero-divisor graph Γ ( S ) . We use these sub-semigroups to study the correspondence between zero-divisor semigroups and zero-divisor graphs. In particular, we discover a class of sub-semigroups of reduced semigroups and we study properties of sub-semigroups of finite or infinite semilattices with the least element. As an application, we provide a characterization of the graphs which are zero-divisor graphs of Boolean rings. We also study how local property of Γ ( S ) affects global property of the semigroup S, and we discover some interesting applications. In particular, we find that no finite or infinite two-star graph has a corresponding nil semigroup.

Semigroups with the ideal retraction property

In this paper we investigate the structure of semigroups with the ideal retraction property i.e., semigroups which are not simple and have the property that each ideal is a homomorphic retract of the semigroup. We present examples to show that the ideal retraction property is neither hereditary nor productive. That this property is preserved by homomorphisms is established for some classes of semigroups, but the general question remains open. The classes of semigroups investigated in this paper are separative semigroups, ideal semigroups, semilattices, cyclic semigroups, nil semigroups, and Clifford semigroups. It is established that a semigroup with zero 0 which is expressible as a direct sum of each ideal and a dual ideal (complement with 0 adjoined) has the ideal retraction property. The converse holds for ideal semigroups, and an example is presented which demonstrates that the converse does not hold in general.

Nil semigroups of endomorphism rings of modules with chain conditions

Nil semigroups of endomorphism rings of modules with chain conditions

Subdirectly dominated semigroup varieties

A variety $ \Bbb V $ is subdirectly dominated if every algebra A in $ \Bbb V $ is embeddable into a subdirectly irreducible algebra belonging to $ \Bbb V $ . We characterize the subdirectly dominated semigroup varieties of nil semigroups, of commutative semigroups, and of completely regular semigroups.

Almeida's Generalized Variety Problem

Let Com denote the variety of all commutative semigroups, let Niℓ denote the generalized variety of all nil semigroups, and let N denote the generalized variety of all nilpotent semigroups. For any class W of semigroups let L(W) denote the lattice of all varieties contained in W, and let G(W) denote the lattice of all generalized varieties contained in W. Almeida has shown that the map ϕ: L(Niℓ∩Com)∪{Niℓ∩Com}→G(N∩Com) given by Wϕ=W∩N is an isomorphism, and asked whether the extension of this map to L(Niℓ)∪{Niℓ} is also an isomorphism. In this article a negative answer is given: two varieties U,V∈L(Niℓ) are defined and used to show that the map is not injective. In the process, congruence classes of the fully invariant congruences on the free semigroup on any countable setXwith |X|≥3 which correspond to U and V are described which are denumerable and contain words of unbounded length.

The lattice of congruences on direct products of cyclic semigroups and certain other semigroups

SynopsisLet W be a semigroup with W\W2 non-empty, such that if ρ is a congruence on W with xpy for all x, y= W\W2, then zpw for all z, w= W2. We prove that the lattice of congruences on W is directly indecomposable, and conclude that a direct product of cyclic semigroups, with at least two non-group direct factors, has a directly indecomposable lattice of congruences. We find that the lattice of congruences on a direct product S1×S2×V of two non-trivial cyclic semigroups S1 and S2, one not being a group, and any other semigroup V, is not lower semimodular, and hence, not modular. We then prove that any finite ideal extension of a group by a nil semigroup has an upper semimodular lattice of congruences, and conclude that a finite direct product of finite cyclic semigroups has an upper semimodular lattice of congruences.

Cancellative and archimedean ideal-extensions by commutative nil semigroups

Cancellative and archimedean ideal-extensions by commutative nil semigroups

A remark on a paper of F. A. Szász

The aim of this short note is to discuss Szisz's paper [1] on rings with almost nil adjoint semigroups and to make some remarks about it. His results should be read as a characterization of rings with almost nit, rather than almost nil adjoint semigroups as stated. All his propositions could be simplified considerably once the word adjoint is dropped. Towards this we state the following result appearing in JACOBSO~ [2].

Semigroups satisfying (xy)m = xmym

In this paper we characterize Archimedean semigroups with idempotents satisfying (xy)m = xmym as exactly those semigroups which are a retract extension of a completely simple semigroup satisfying (xy)m = xmym by a nil semigroup satisfying (xy)m = xmym. Regular semigroups satisfying (xy)2 = x2y2 are exactly those semigroups which are a spined product of a band and a semigroup which is a semilattice of Abelian groups. A semigroup which is a nil extension of a regular semigroup satisfies (xy)2 = x2y2 if and only if it is a retract extension of a regular semigroup satisfying (xy)2 = x2y2 by a nil semigroup satisfying (xy)2 = x2y2

Nildemi-groupes totalement ordonnés

Nildemi-groupes totalement ordonnés

Rings satisfying polynomial constraints

The following theorem is proved: Suppose R is an associative ring and suppose J is the Jacobson radical of R. Suppose that for all x1, * *, xn in R, there exists a word wx... (x1, * * x), depending on x1, * *, x,,, in which at least one xi (i varies) is missing, and such that xi ... x, = w ,... .n(x1, , x). Then J is a nil ring of bounded index and R/J is finite. It is further proved that a commutative nil semigroup satisfies the above identity if and only if it is nilpotent. In this paper, we investigate the structure of an associative ring R with the property that, for all xl, * * * , x, in R, xl ... x = wxi.... xn(xl, * * *, Xn), where w is a word, depending on xl, * * *, xn, and where some xi (i varies) is missing in w. For such a ring R, we prove the Jacobson radical J is a nil ring of bounded index. We also show that R/J is finite. Finally, we show that a commutative nil semigroup satisfies the above identity if and only if it is nilpotent. In preparation for the proofs of the main results, we first introduce DEFINITION 1. Let n be a positive integer, and let S be a semigroup. We say that S is a fln-semigroup if, for all xl, * * *, xn in S, there exists a word w .(x, *1 , xn) consisting of a product of the xi's with some x1 (j varies) missing, such that xl ... xn=wxl....xn((xl, * * *, xn). A ring R is called a fln-ring if its multiplicative semigroup is a fln-semigroup. LEMMA 1. Let S be a finite semigroup or a nilpotent semigroup. Then S is a fln-semigroup, for some n. PROOF. First, suppose S is finite, of order n, and suppose xl, Xn+1 are any elements of S. Then xi=xj for some i>j, and hence X1 ... Xn+1 = X1 ... Xi ... Xi-lXjXi+l Xn+1 = W(X1, . . , Xi-1 Xi+1 , Xn+1) Thus S is a fln+l-semigroup. Next, suppose S is nilpotent, say, Sm= (0). Presented to the Society, January 18, 1972; received by the editors August 10, 1971 and, in revised form, October 6, 1971. AMS (MOS) subject classifications (1970). Primary 16A38, 16A48; Secondary 20M10. The author was supported by a National Science Foundation Graduate Fellowship. @ American Mathematical Society 1973

Translations of commutative unique factorization semigroups

Every commutative semigroup is a semilattice of archimedean subsemigroups. Given a semilattice Y and a collection of commutative semigroups S indexed by Y, the converse problem of constructing every commutative semigroup which is a semilattice (Y) of the semigroups S remains unsolved except in certain cases. If each S is an abelian group the result follows as a special case of Clifford's Theorem (see [33, Thm. 4.11). The authors [13 have given a solution for any semilattice where each S is assumed to be cyclic, and more recently, Tamura has solved the problem for exclusive semilattices of exclusive semigroups [63 For a characterization of the general semilattice decomposition when each Se is weakly reductive, see Theorem 7.8.13 of [41. An element x of a commutative semigroup S is said to be prime if x ~ S 2 . S is said to have unique factorization if each nonzero element of S can be written uniquely as a product of primes in the usual sense. A commutative semigroup has unique factorization if and only if it is free or a Rees quotient of a free commutative semigroup (E2~, Thm. i). Commutative nil unique factorization semigroups were used in the characterization of arbitrary commutative nil semigroups in [11 (see [53 for an alternate characterization). Since commutative unique factorization semigroups arise naturally in the general theory of commutative semigroups, it seems of interest

Nil Semi-Groups of Rings with a Polynomial Identity

The basic properties of associative rings R satisfying a polynomial identity p[x1…, xn] = 0 were obtained under the assumptions that the ring was an algebra [e.g., [4] Ch. X], or with rather strong restrictions on the ring of operators ([1]). But it is desirable to have these properties for arbitrary rings, and the present paper is the first of an attempt in this direction. The problem is almost trivial for prime or semi-prime rings but quite difficult in arbitrary rings. The known proofs for algebras have to be modified and in some cases new proofs have to be obtained as the existing proofs fail to exploit the known structure. In the present paper we extend the results of [1] on the nil subalgebras of a ring with an identity for arbitrary multiplicative nil semi-groups of the ring and for arbitrary rings.