International Journal of Algebra and Computation

It is proved that the periodic point submonoid of a free inverse monoid endomorphism is always finitely generated. Using Chomsky's hierarchy of languages, we prove that the fixed point submonoid of an endomorphism of a free inverse monoid can be represented by a context-sensitive language but, in general, it cannot be represented by a context-free language.

Convex subsets of affine spaces over the field of real numbers are described by so-called barycentric algebras. In this paper, we discuss extensions of the geometric and algebraic definitions of a convex set to the case of more general coefficient rings. In particular, we show that the principal ideal subdomains of the reals provide a good framework for such a generalization. Since the closed intervals of these subdomains play an essential role, we provide a detailed analysis of certain cases, and discuss differences from the "classical" intervals of the reals. We introduce a new concept of an algebraic closure of "geometric" convex subsets of affine spaces over the subdomains in question, and investigate their properties. We show that this closure provides a purely algebraic description of topological closures of geometric generalized convex sets. Our closure corresponds to one instance of the very general closure introduced in an earlier paper of the authors. The approach used in this paper allows to extend some results from that paper. Moreover, it provides a very simple description of the closure, with concise proofs of existence and uniqueness.

A group G is a vGBS group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We describe the compatibility JSJ decomposition over abelian groups. We prove that in general this decomposition is not algorithmically computable.

We study the nilpotency degree of a relatively free finitely generated associative algebra with the identity xⁿ = 0 over a finite field 𝔽 with q elements. In the case of q ≥ n the nilpotency degree is proven to be the same as in the case of an infinite field of the same characteristic. In the case of q = n - 1 it is shown that the nilpotency degree differs from the nilpotency degree for an infinite field of the same characteristic by at most one. The nilpotency degree is explicitly computed for n = 3.

An open problem in the theory of Moufang loops is to classify those loops which are minimally nonassociative, that is, loops which are nonassociative but where all proper subloops are associative. A related question is to classify all integers n for which a minimally nonassociative loop exists. In [Possible orders of nonassociative Moufang loops, Comment. Math. Univ. Carolin.41(2) (2000) 237–244], O. Chein and the third author showed that a minimal nonassociative Moufang loop of order 2q³ can be constructed by using a non-abelian group of order q³. In [Moufang loops of odd order pq³, J. Algebra235 (2001) 66–93], the third author also proved that for odd primes p < q, a nonassociative Moufang loop of order pq³ exists if and only if q ≡ 1 (mod p). Here we complete the classification of minimally nonassociative Moufang loops of order pq³ for primes p < q.

Let R = ⨁_α∈Γ R_α be a (Γ-)graded integral domain and let H be the multiplicatively closed set of nonzero homogeneous elements of R. In this paper, we introduce the concepts of graded almost GCD-domains (graded AGCD-domain) and graded almost Prüfer v-multiplication domains (graded APvMD). Among other things, we show that if R is integrally closed, then (1) H is an almost lcm splitting set of R if and only if R is a graded AGCD-domain and (2) R is a graded APvMD if and only if R is a PvMD. We also give an example of a (non-integrally closed) graded AGCD-domain (respectively, graded APvMD) that is not an almost GCD-domain (respectively, almost Prüfer v-multiplication domain.

We show that, like in the case of algebras over fields, the study of multilinear polynomial identities of unitary rings can be reduced to the study of proper polynomial identities. In particular, the factors of series of ℤS_n-submodules in the ℤS_n-modules of multilinear polynomial functions can be derived by the analog of Young's (or Pieri's) rule from the factors of series in the corresponding ℤS_n-modules of proper polynomial functions. As an application, we calculate the codimensions and a basis of multilinear polynomial identities of unitary rings of upper triangular 2 × 2 matrices and infinitely generated Grassmann algebras over unitary rings. In addition, we calculate the factors of series of ℤS_n-submodules for these algebras. Also we establish relations between codimensions of rings and codimensions of algebras and show that the analog of Amitsur's conjecture holds in all torsion-free rings, and all torsion-free rings with 1 satisfy the analog of Regev's conjecture.