Left Reductive Congruences on Semigroups

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

We consider semigroups such that the universal left congruence omega ^{ell } is finitely generated. Certainly a left noetherian semigroup, that is, one in which all left congruences are finitely generated, satisfies our condition. In the case of a monoid the condition that omega ^{ell } is finitely generated is equivalent to a number of pre-existing notions. In particular, a monoid satisfies the homological finiteness condition of being of type left-{text {FP}}_1 exactly when omega ^{ell } is finitely generated.Our investigations enable us to classify those semigroups such that omega ^{ell } is finitely generated that lie in certain important classes, such as strong semilattices of semigroups, inverse semigroups, Rees matrix semigroups (over semigroups) and completely regular semigroups. We consider closure properties for the class of semigroups such that omega ^{ell } is finitely generated, including under morphic image, direct product, semi-direct product, free product and 0-direct union. Our work was inspired by the stronger condition, stated for monoids in the work of White, of being pseudo-finite. Where appropriate, we specialise our investigations to pseudo-finite semigroups and monoids. In particular, we answer a question of Dales and White concerning the nature of pseudo-finite monoids.

New precedence results are obtained for semigroups without zero, which describe the first row in detail and provide effective comparisons with the next few rows. These results can be used to further sharpen existing algorithms for the computation of finite commutative semigroups.

For any commutative semigroup S and positive integer m the power function f: S rightarrow S defined by f(x) = x^m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

We investigate the intersection problem for finite semigroups, which asks for a given set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. We introduce compressibility measures as a useful tool to classify the intersection problem for certain classes of finite semigroups into circuit complexity classes and Turing machine complexity classes. Using this framework, we obtain a new and simple proof that for groups and commutative semigroups, the problem is contained in \(\mathsf {NP}\). We uncover certain structural and non-structural properties determining the complexity of the intersection problem for varieties of semigroups containing only trivial submonoids. More specifically, we prove \(\mathsf {NP}\)-hardness for classes of semigroups having a property called unbounded order and for the class of all nilpotent semigroups of bounded order. On the contrary, we show that bounded order and commutativity imply containment in the circuit complexity class \(\mathsf {qAC}^k\) (for some \(k \in \mathbb {N}\)) and decidability in quasi-polynomial time. We also establish connections to the monoid variant of the problem.

Alomair et al. (J Math Cryptol 4(2):121–148, 2010, Lemma 3.1) noticed the following result which seems not to appear previously explicitly in the literature: Given a nonzero \(a \in \mathbb{Z}_{n}\), the ring of residues modulo n, such that gcd(a, n) = d | b, not only there exists an element \(x \in \mathbb{Z}_{n}\) such that \(x \cdot a \equiv b\pmod n\), but that there even exists an invertible element \(x \in \mathbb{Z}_{n}^{{\ast}}\) such that \(x \cdot a \equiv b\pmod n\). Their sufficient and necessary condition for this says that gcd(b∕d, n∕d) = 1 with d as above.A typical structure result on finite commutative semigroup says that the multiplicative semigroup of \(\mathbb{Z}_{n}\) decomposes into the so-called maximal subsemigroups belonging to the idempotents of \(\mathbb{Z}_{n}\). Each such semigroup contains a maximal subgroup having for its identity the corresponding idempotent. In general this subgroup is a proper subset of the maximal subsemigroup containing it. However, the group of elements of \(\mathbb{Z}_{n}\) coprime to n is an example of the case when this maximal subsemigroup and the maximal subgroup coincide (both evidently belonging to the idempotent 1).In what follows we prove that if a congruence \(x \cdot a \equiv b\pmod n\) is solvable there always exists a solution in the maximal semigroup belonging to the idempotent given by the divisor δ = gcd(b∕d, n∕d) and if δ is a unitary divisor of n then there even exists a solution in the maximal subgroup belonging to the idempotent given by δ.

Nilpotent semigroups in the sense of Mal’cev are defined by semigroup identities. Finite nilpotent semigroups constitute a pseudovariety, MN, which has finite rank. The semigroup identities that define nilpotent semigroups lead us to define strongly Mal’cev nilpotent semigroups. Finite strongly Mal’cev nilpotent semigroups constitute a non-finite rank pseudovariety, SMN. The pseudovariety SMN is strictly contained in the pseudovariety MN, but all finite nilpotent groups are in SMN. We show that the pseudovariety MN is the intersection of the pseudovariety BGnil with a pseudovariety defined by a κ-identity. We further compare the pseudovarieties MN and SMN with the Mal’cev product 𝖩ⓜ𝖦nill.

Right zero unions of N-semigroups

The intersection problem for finite semigroups asks, given a set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. In previous work, it was shown that is problem is [Formula: see text]-complete. We introduce compressibility measures as a useful tool to classify the complexity of the intersection problem for certain classes of finite semigroups. Using this framework, we obtain a new and simple proof that for groups and for commutative semigroups, the problem (as well as the variant where the languages are represented by finite automata) is contained in [Formula: see text]. We uncover certain structural and non-structural properties determining the complexity of the intersection problem for varieties of semigroups containing only trivial submonoids. More specifically, we prove [Formula: see text]-hardness for classes of semigroups having a property called unbounded order and for the class of all nilpotent semigroups of bounded order. On the contrary, we show that bounded order and commutativity imply decidability in poly-logarithmic time on alternating random-access Turing machines with a single alternation. We also establish connections to the monoid variant of the problem.

The homological theory of rings and modules is an important branch of algebra. It provided answers to numerous questions of the theory of rings. Along with the homological theory, another theory started to develop, also under significant influence of the theory of rings, which is the homological theory of universal algebras, and, in particular, of semigroups and acts over them. This theory analyses such notions as injective and projective acts over semigroups, injective hulls and projective covers. As in the case of rings and modules, the injective hull exists for every act, while the projective cover sometimes does not. In 1967 P. Berthiaume proved the existence of injective hulls of an arbitrary act over a semigroup (without the assumption of the presence of an identity in the semigroup). J. Isbell studied monoids (i.e. semigroups with an identity) over which every act has a projective cover. L. A. Skornyakov developed a homological theory of monoids. Many results of that theory were mentioned in the known monograph by M. Kilp, U. Knauer, A. V. Mikhalev. For semigroups of a relatively simple structure the results of the homological theory can be significantly refined. For example, in 2012 G. Moghaddasi described injective acts and built injective hulls of acts over a left zero semigroup assuming the separability of the act. I. B. Kozhukhov and A. P. Haliullina described injective and projective acts over groups and right zero semigroups, built injective hulls and projective covers of acts over such semigroups. For acts over a left zero semigroup the condition of separability of acts was removed. An important class of semigroups containing groups, left and right zero semigroups, rectangular bands is the class of completely simple semigroups, as well as the broader class of completely 0-simple semigroups. In 2000 A. Yu. Avdeyev and I. B. Kozhukhov described all acts over completely simple semigroups and acts with zero over completely 0-simple semigroups. It triggered further reasearch of acts over such semigroups. I. B. Kozhuhov and A. O. Petrikov described injective and projective acts over completely simple semigroups, thereby generalising the results of I. B. Kozhuhov and A. R. Khaliullina, and also the work of G. Mogaddasi. They built injective hulls and projective covers of acts over such semigroups. In this paper the above-mentioned results concerning acts over completely simple semigroups were generalized to acts with zero over completely 0-simple semigroups. In particular, the necessary and sufficient conditions of injectivity and projectivity of an act with zero over an arbitrary completely 0-simple semigroup were found, injective hulls and projective covers of arbitrary acts with zero over such semigroups were built. It was established that a projective act over an arbitrary completely 0-simple semigroup is exactly a 0-coproduct of a free act and acts isomorphic to a 0-minimal right ideal of the semigroup (considered as a right act).

We give a description of finite semigroups S that are minimal for not being Malcev nilpotent, i.e. every proper subsemigroup and every proper Rees factor semigroup is Malcev nilpotent but S is not. For groups this question was considered by Schmidt.

The free envelope of a finite commutative semigroup was defined by Grillet [Trans. Amer. Math. Soc.

Matrix representations of finite commutative semigroups

In this article we give a criterion for the semigroup algebra of a finite commutative semigroup over a field to be Frobenius.

The generalized firing squad synchronization problem (GFSSP) is the well-known firing squad synchronization problem extended to arbitrarily connected networks of finite automata. When the transmission delays associated with the links of a network are allowed to be arbitrary nonnegative integers, the problem is called GFSSP-NUD (GFSSP with nonuniform delays). A solution of GFSSP-NUD is given for the first time. The solution is independent of the structure of the network and the actual delays of the links. The firing time of the solution is bounded by O( Delta /sup 3/+ tau /sub max/), where tau /sub max/ is the maximum transmission delay of any single link and Delta is the maximum transmission delay between the general and any other node of a given network. Extensions of GFSSP and GFSSP-NUD to networks with more than one general are presented. >