Narrow Results

The purpose of this paper is to compute the nonzero semigroup determinant of the class of finite semigroups in which every two idempotents commute. This class strictly contains the class of finite semigroups that have central idempotents and the class of finite inverse semigroups. This computation holds significance in the context of the extension of the MacWilliams theorem for codes over semigroup algebras.

We consider algebras over a field with generators x₁,x₂,…,x_n subject to square-free relations x_ix_j=x_kx_l in which every product x_px_q, p≠q, appears in one of the relations. The work of Gateva-Ivanova and Van den Bergh, motivated in particular by the study of set theoretic solutions of the Yang–Baxter equation, provided an important class of such algebras. In this special case the presentation satisfies the so-called cyclic condition that became an essential combinatorial tool in proving that these algebras share many strong ring theoretic properties of polynomial algebras in commuting variables. In this paper we describe the structure of algebras on four generators satisfying the cyclic condition. The emphasis is on some new unexpected features, not present in the motivating special classes.

Let A be a finitely generated commutative algebra over a field K with a presentation A = K 〈X₁,…, X_n | R〉, where R is a set of monomial relations in the generators X₁,…, X_n. So A = K[S], the semigroup algebra of the monoid S = 〈X₁,…, X_n | R〉. We characterize, purely in terms of the defining relations, when A is an integrally closed domain, provided R contains at most two relations. Also the class group of such algebras A is calculated.

The class of finitely presented algebras over a field $K$ with a set of generators $a_1,\dots ,a_n$ and defined by homogeneous relations of the form $a_1{a}_2\cdots a_n=a_{\sigma (1)}{a}_{\sigma (2)}\cdots a_{\sigma (n)}$, where $\sigma$ runs through a subset $H$ of the symmetric group ${Sym}_n$ of degree $n$, is investigated. Groups $H$ in which the cyclic group $〈(1,2,\dots ,n)〉$ is a normal subgroup of index $2$ are considered. Certain representations by permutations of the dihedral and semidihedral groups belong to this class of groups. A normal form for the elements of the underlying monoid $S_n(H)$ with the same presentation as the algebra is obtained. Properties of the algebra are derived, it follows that it is an automaton algebra in the sense of Ufnarovskij. The universal group $G_n$ of $S_n(H)$ is a unique product group, and it is the central localization of a cancellative subsemigroup of $S_n(H)$. This, together with previously obtained results on such semigroups and algebras, is used to show that the algebra $K[S_n(H)]$ is semiprimitive.

Let $A=ℝ〈a_1,\dots ,a_n〉$ and $B=ℝ〈b_1,\dots ,b_n〉$ be the semigroup rings spanned on the right zero semigroup ${\mathrm{\text{RZ}}}_n=\{a_1,\dots ,a_n\}$, and on the left zero semigroup ${\mathrm{\text{LZ}}}_n=\{b_1,\dots ,b_n\}$, respectively, together with the identity element $1$. We suggest a closed formula solving the equation $w=log(e^u{e}^v)$ which is the evolution of the Campbell–Baker–Hausdorff formula given by the Hausdorff series

Let H be a subgroup of Sym_n, the symmetric group of degree n. For a fixed integer l ≥ 2, the group G presented with generators x₁, x₂,…,x_n and with relations x_i1x_i2⋯x_il = x_σ(i1) x_σ(i2)⋯x_σ(il), where σ runs through H, is considered. It is shown that G has a free subgroup of finite index. For a field K, properties of the algebra K[G] are derived. In particular, the Jacobson radical 𝒥(K[G]) is always nilpotent, and in many cases the algebra K[G] is semiprimitive. Results on the growth and the Gelfand–Kirillov dimension of K[G] are given. Further properties of the semigroup S and the semigroup algebra K[S] with the same presentation are obtained, in case S is cancellative. The Jacobson radical is nilpotent in this case as well, and sufficient conditions for the algebra to be semiprimitive are given.

A finite Gröbner-Shirshov basis is constructed for the plactic algebra of rank 3 over a field K. It is also shown that plactic algebras of rank exceeding 3 do not have finite Gröbner-Shirshov bases associated to the natural degree-lexicographic ordering on the corresponding free algebra. The latter is in contrast with the case of a strongly related class of algebras, called Chinese algebras.