Narrow Results

Any semigroup S can be embedded into a semigroup, denoted by ΨS, having some remarkable properties. For general semigroups there is a close relationship between local submonoids of S and of ΨS. For a number of usual semigroup properties , we prove that S and ΨS simultaneously satisfy or not. For a regular semigroup S, the relationship of S and ΨS is even closer, especially regarding the natural partial order and Green's relations; in addition, every element of ΨS is a product of at most four idempotents. For completely regular semigroups S, the relationship of S and ΨS is still closer. On the lattice of varieties of completely regular semigroups regarded as algebras with multiplication and inversion, by means of ΨS, we define an operator, denoted by Ψ. We compare Ψ with some of the standard operators on and evaluate it on a small sublattice of .

Let G be a group and let w = w(x₁, x₂,…, x_n) be a word in the absolutely free group F_n on free variables x₁, x₂,…, x_n. The set S⁽ⁿ⁾(G) of all words w such that the equality w(g_σ1, g_σ2,…, g_σn) = w(g₁, g₂,…, g_n) holds for all g₁, g₂,…, g_n∈G and all permutations σ ∈ S_n is a subgroup of F_n, called the subgroup of n-symmetric words for G. In this paper, the groups S⁽²⁾(D_p) and S⁽³⁾(D_p) for dihedral groups D_p are determined, where p > 3 is a prime. In particular, it turns out that the groups S⁽³⁾(D_p) are not abelian.

Assume that is the variety of bands, and the identities of involve n variables (n ≥ 2). In this paper, we show that is -testable and is not -testable.

Let 𝔽_q stand for the finite field of odd characteristic p with q elements (q = pⁿ, n ∈ ℕ) and ${\mathbb{F}}_q^{*}$ denote the set of all the nonzero elements of 𝔽_q. Let m and t be positive integers. By using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over ${\mathbb{F}}_q:{\displaystyle \sum _{j=0}^{t-1}{\displaystyle \sum _{i=1}^{r_{j+1}-r_j}{a}_{k,r_j+i}{x}_1{}^{e_{r_j+i,1}^{\left(k\right)}}\dots x_{n_{j+1}}^{e_{r_j+i,n_{j+1}}^{\left(k\right)}}=b_{k,}}}k=1,\dots ,m,$ where the integers t > 0, r₀ = 0 < r₁ < r₂ < ⋯ < r_t, 1 ≤ n₁ < n₂ <, ⋯ < n_t and 0 ≤ j ≤ t − 1, b_k ∊ 𝔽_q, a_k,i ∊ ${\mathbb{F}}_q^{*}$ (k = 1, …, m, i = 1, …, r_t), and the exponent of each variable is a positive integer. Further, under some natural conditions, we arrive at an explicit formula for the number of 𝔽_q-rational points on the above variety. It extends the results obtained previously by Wolfmann, Sun, Wang, Hong et al. Our result gives a partial answer to an open problem raised in [The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory156 (2015) 135–153].

In this paper, we give a sufficient condition under which an involution monoid generates a variety with continuum many subvarieties. According to this result, several involution $J$-trivial monoids are shown to generate varieties with continuum many subvarieties. These examples include Rees quotients of free involution monoids, Lee monoids with involution, and Straubing monoids with involution.

A limit variety is a variety that is minimal with respect to being non-finitely based. Since the turn of the millennium, much attention has been given to the classification of limit varieties of aperiodic monoids. Seven explicit examples have so far been found, and the task of locating other examples has recently been reduced to two subproblems, one of which is concerned with monoids that satisfy the identity $xsxt\approx xsxtx$. We provide a complete solution to this subproblem by showing that there are precisely two limit varieties that satisfy this identity. One of them turns out to be the first example having infinitely many subvarieties. It is also deduced that the variety generated by any monoid of order 5 or less contains at most countably many subvarieties.