We consider GL -dimers of triangulations of regular convex -gons, which give rise to a dimer model with boundary and a dimer algebra Λ . Let be the sum of the idempotents of all the boundary vertices, and the associated boundary algebra. In this article we show that given two different triangulations and of the -gon, the boundary algebras are isomorphic, i.e.

We decompose certain nilpotent expanded groups into a direct product such that the additive group of each factor is either a -group or torsion-free.

Let be a transfer Krull monoid over a finite abelian group (for example, rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains). Then each nonunit ∈ can be written as a product of irreducible elements, say , and the number of factors is called the length of the factorization. The set () of all possible factorization lengths is the set of lengths of . It is classical that the system () = {()∣∈} of all sets of lengths depends only on the group , and a standing conjecture states that conversely the system () is characteristic for the group . Let be a further transfer Krull monoid over a finite abelian group and suppose that () = ( ). We prove that, if with ≤-3 or (≥-1≥2 and is a prime power), then and are isomorphic.

Rota-Baxter operators of weight 1 on are in bijective correspondence to post-Lie algebra structures on pairs , where is complete. We use such Rota-Baxter operators to study the existence and classification of post-Lie algebra structures on pairs of Lie algebras , where is semisimple. We show that for semisimple and , with or simple, the existence of a post-Lie algebra structure on such a pair implies that and are isomorphic, and hence both simple. If is semisimple, but is not, it becomes much harder to classify post-Lie algebra structures on , or even to determine the Lie algebras which can arise. Here only the case was studied. In this paper, we determine all Lie algebras such that there exists a post-Lie algebra structure on with .

For a partial differential field , we show that the triviality of the first differential Galois cohomology of every linear differential algebraic group over is equivalent to being algebraically, Picard-Vessiot, and linearly differentially closed. This cohomological triviality condition is also known to be equivalent to the uniqueness up to an isomorphism of a Picard-Vessiot extension of a linear differential equation with parameters.

In this article, we investigate the concept of radical factorization with respect to finitary ideal systems of cancellative monoids. We present new characterizations for -almost Dedekind -SP-monoids and provide specific descriptions of -almost Dedekind -SP-monoids and -SP-monoids. We show that a monoid is a -SP-monoid if and only if the radical of every nontrivial principal ideal is -invertible. We characterize when the monoid ring is a -SP-domain and describe when the *-Nagata ring is an SP-domain for a star operation * of finite type.

In this paper, we determine all the Rota-Baxter operators of weight zero on semigroup algebras of order two and three with the help of computer algebra. We determine the matrices for these Rota-Baxter operators by directly solving the defining equations of the operators. We also produce a Mathematica procedure to predict and verify these solutions.

An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one (essentially different) factorization into irreducibles. In the case of the ring of integer-valued polynomials on a principal ideal domain with quotient field , we give an easy to verify graph-theoretic sufficient condition for an element to be absolutely irreducible and show a partial converse: the condition is necessary and sufficient for polynomials with square-free denominator.

A commutative integral domain is primary if and only if it is one-dimensional and local. A domain is strongly primary if and only if it is local and each nonzero principal ideal contains a power of the maximal ideal. Hence, one-dimensional local Mori domains are strongly primary. We prove among other results that if R is a domain such that the conductor vanishes, then is finite; that is, there exists a positive integer k such that each nonzero nonunit of R is a product of at most k irreducible elements. Using this result, we obtain that every strongly primary domain is locally tame, and that a domain R is globally tame if and only if In particular, we answer Problem 38 of the open problem list by Cahen et al. in the affirmative. Many of our results are formulated for monoids.

Mazurov asked whether a group of exponent dividing 12, which is generated by , and subject to the relations has order at most 12. We show that if such a group is finite, then the answer is yes.

A commutative ring is stable if every non-zero ideal of is projective over its ring of endomorphisms. Motivated by a paper of Bass in the 1960s, stable rings have received wide attention in the literature ever since then. Much is known on the algebraic structure of stable rings and on the relationship of stability with other algebraic properties such as divisoriality and the 2-generator property. In the present paper, we study the arithmetic of stable integral domains, with a focus on arithmetic properties of semigroups of ideals of stable orders in Dedekind domains.

We study the existence of post-Lie algebra structures on pairs of Lie algebras , where one of the algebras is perfect non-semisimple, and the other one is abelian, nilpotent non-abelian, solvable non-nilpotent, simple, semisimple non-simple, reductive non-semisimple or complete non-perfect. We prove several nonexistence results, but also provide examples in some cases for the existence of a post-Lie algebra structure. Among other results we show that there is no post-Lie algebra structure on , where is perfect non-semisimple, and is . We also show that there is no post-Lie algebra structure on , where is perfect and is reductive with a 1-dimensional center.