Narrow Results

Dyadic rationals are rationals whose denominator is a power of 2. Dyadic triangles and dyadic polygons are, respectively, defined as the intersections with the dyadic plane of a triangle or polygon in the real plane whose vertices lie in the dyadic plane. The one-dimensional analogs are dyadic intervals. Algebraically, dyadic polygons carry the structure of a commutative, entropic and idempotent algebra under the binary operation of arithmetic mean. In this paper, dyadic intervals and triangles are classified to within affine or algebraic isomorphism, and dyadic polygons are shown to be finitely generated as algebras. The auxiliary results include a form of Pythagoras' theorem for dyadic affine geometry.

A semigroup S is of the type in the class of the title if S has a congruence ρ such that S/ρ is a normal band (i.e. satisfies the identities x² = x and axya = ayxa) and all ρ-classes are commutative cancellative semigroups. We consider semigroups S with such a congruence first for completely regular semigroups, then characterize the general case in several ways, including some special cases. When S is an order in a normal band of abelian groups Q, we study the restrictions of Green's relations on Q to S. The paper concludes with the discussion of a free semigroup in the title on two generators.

We give a simple characterization for a nonassociative algebra , having characteristic ≠2, to be commutative. Namely, is commutative if and only if it is flexible with a commuting set of generators. A counterexample shows that characteristic ≠2 is necessary. Both the characterization and the counterexample were discovered using the computer algebra system in [2].

Anti-elementarity is a strong way of ensuring that a class of structures, in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form ${ℒ}_{\infty \lambda }$. We prove that many naturally defined classes are anti-elementary, including the following:

• the class of all lattices of finitely generated convex $\ell$-subgroups of members of any class of $\ell$-groups containing all Archimedean $\ell$-groups;
• the class of all semilattices of finitely generated $\ell$-ideals of members of any nontrivial quasivariety of $\ell$-groups;
• the class of all Stone duals of spectra of MV-algebras — this yields a negative solution to the MV-spectrum Problem;
• the class of all semilattices of finitely generated two-sided ideals of rings;
• the class of all semilattices of finitely generated submodules of modules;
• the class of all monoids encoding the nonstable K₀-theory of von Neumann regular rings, respectively, C*-algebras of real rank zero;
• (assuming arbitrarily large Erdős cardinals) the class of all coordinatizable sectionally complemented modular lattices with a large $4$-frame.

The main underlying principle is that under quite general conditions, for a functor $\mathrm{\Phi }:{𝒜}\to {ℬ}$, if there exists a noncommutative diagram $\overrightarrow{D}$ of ${𝒜}$, indexed by a common sort of poset called an almost join-semilattice, such that

• $\mathrm{\Phi }{\overrightarrow{D}}^I$ is a commutative diagram for every set $I$,
• $\mathrm{\Phi }\overrightarrow{D}\ncong \mathrm{\Phi }\overrightarrow{X}$ for any commutative diagram $\overrightarrow{X}$ in ${𝒜}$,

then the range of $\mathrm{\Phi }$ is anti-elementary.

We characterize the structure of finite unitary rings $R$ in which every proper subring is commutative and show that $R$ is a commutative ring.

The aim of this paper is to study the structure of irreducible modules in the variety ${ℳ}$ of commutative power-associative nilalgebras of nilindex $\le 4$. If $A\in {ℳ}$ with dimension at most 5, then we prove that $A^2$ is contained in the annihilator of every irreducible $A$-module in the variety ${ℳ}$. Also, we consider the enveloping algebra of an algebra $A$ in the variety ${ℳ}$ and we obtain a new example of a commutative power-associative non-nilpotent nilalgebra of dimension 9.

In this paper, the Composition-Diamond lemma for commutative algebras with multiple operators is established. As applications, the Gröbner–Shirshov bases and linear bases of free commutative Rota–Baxter algebra, free commutative λ-differential algebra and free commutative λ-differential Rota–Baxter algebra are given, respectively. Consequently, these three free algebras are constructed directly by commutative Ω-words.

Given a group $G$, we study right and left zeros, idempotents, the minimal ideal, left cancelable and right cancelable elements of the semigroup $N_k(G)$ of $k$-linked upfamilies and characterize groups $G$ whose extensions $N_k(G)$ are commutative. We finish the paper with the complete description of the structure of the semigroups $N_k(G)$ for all groups $G$ of cardinality $\left|G\right|\le 4$.

In this paper, we find sufficient conditions for some commutative varieties of semigroups to be closed under dominions.

A Kähler graph consists of two kinds of graphs having the same set of vertices. As candidates of model Kähler graphs which correspond to homogeneous spaces admitting magnetic fields, vertex transitive normal Kähler graphs are proposed in [5]. To give many examples of such Kähler graphs, we define some product operations. In this note, we restrict ourselves to the product Kähler graphs whose first kind graphs are of tensor product type, and study whether they are connected, are bipartite, are vertex-transitive and are normal.