Commutative Nilsemigroups Research Articles

Finite Structurally Uniform Groups and Commutative Nilsemigroups

Let S be a finite semigroup. By Sub(S) we denote the lattice of all its subsemigroups. If A ∈ Sub(S), then by h(A) we denote the height of a subsemigroup A in the lattice Sub(S). A semigroup S is called structurally uniform if, for any A,B ∈ Sub(S) the condition h(A) = h(B) implies that A ≅ B. We present a classification of finite structurally uniform groups and commutative nilsemigroups.

On lattices embeddable into subsemigroup lattices. III: Nilpotent semigroups

We prove that the class of the lattices embeddable into subsemigroup lattices of n-nilpotent semigroups is a finitely based variety for all n < ω. Repnitskiĭ showed that each lattice embeds into the subsemigroup lattice of a commutative nilsemigroup of index 2. In this proof he used a result of Bredikhin and Schein which states that each lattice embeds into the suborder lattices of an appropriate order. We give a direct proof of the Repnitskiĭ result not appealing to the Bredikhin-Schein theorem, so answering a question in a book by Shevrin and Ovsyannikov.

A construction of commutative nilsemigroups

All commutative nilsemigroups are constructed from congruences on free commutative semigroups.

Commutative Nilsemigroups with Zero Cohomology

The overpath method is used to characterize finite commutative nilmonoids whose second commutative cohomology group vanishes at all Schutzenberger functors. AMS Classification: 20M14, 20M50.

On subsemigroup lattices without non-trivial identities

A latticeL is called discriminating if for any free latticeF and for any finite number of elementsu1,u2, ...,un∈F, there exists a homomorphismf:F→L such thatf(ui)≠f(uj) wheneverui ≠ uj (1≤i, j≤n). In this paper it is proved that the subsemigroup lattice SubS of a commutative semigroupS does not satisfy a non-trivial identity if and only if SubS is discriminating. In particular, in this case every finite projective lattice can be embedded into SubS. It should be noted that the most important examples of semigroups whose subsemigroup lattices satisfy no non-trivial identity and therefore have the discriminating property are the following: the infinite cyclic semigroup, the free semilattice of countable rank, any commutative nilsemigroup which is not nilpotent and so on.

The commutative cohomology of nilsemigroups

Our main result computes the second cohomology group of any finite commutative nilsemigroup with identity adjoined, in terms of its presentation.

Nilsemigroups on trees

We show how to reconstruct finite commutative nilsemigroups from their underlying partially ordered sets, and characterize which finite trees underlie finite commutative nilsemigroups.

A construction of finite commutative nilsemigroups

A construction of finite commutative nilsemigroups

Cancellative and archimedean ideal-extensions by commutative nil semigroups

Cancellative and archimedean ideal-extensions by commutative nil semigroups

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