Introduction to Separable Simplicial Algebras

We investigate core quandles and idempotents in quandle rings of core quandles. We answer several questions on the rank of core quandles and nontrivial idempotents in quandle rings. We also present solutions to two questions raised in a recent paper about non-trivial idempotents in quandle rings $\mathbb{Z}[R_5]$ and $\mathbb{Z}[C_5]$, where $\mathbb{Z}$, $R_5$ and $C_5$ are the ring of integers, the dihedral quandle of order 5, and the commutative quandle of order 5, respectively. We then study units in extended quandle rings of a trivial quandle and the Joyce quandle, and nilpotent elements in extended quandles rings of a trivial quandle, where the ground ring is an integral domain. As a consequence, we show that the quandle ring and the extended quandle ring of a trivial quandle are not nil clean rings. We also explore prime rings and semi-prime rings among quandle rings. We introduce zero-divisor graphs of quandle rings and find an intriguing mirror symmetry among the in-degree and out-degree of the vertices. Moreover, we find a bivariate polynomial in the ring $(\mathbb{Z}_{2n+1}[Q])[X,Y]$ that determines the commutative quandle of order $2n+1$, where $2n+1$ is prime.

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arXiv2026

A lattice order on the complex field

Birkhoff and Pierce asked in 1956 whether the field $\mathbb{C}$ of complex numbers admits a lattice order compatible with addition and multiplication. We answer the question affirmatively. Using a valuation with value group $\mathbb{Q}$, we construct a positive cone $P$ by a transfinite recursion of countable ordinal length. At each stage we extend a lattice order on a subspace while preserving its lattice operations in the whole field and maintaining multiplicative closure of the ambient positive cone. These subspaces eventually exhaust $\mathbb{C}$. The construction depends on choices of a valuation extension, compatible rational powers and vector-space bases. The cone $P$ contains $1$ and satisfies $\{x\in P\setminus\{0\}:x^{-1}\in P\}=\mathbb{Q}_{>0}$. The induced order is not total: $i$ is incomparable with $0$. For a chosen transcendental element $t$ and $i^2=-1$, one may prescribe $0\vee i=(t+i)/2$ and $0\vee(-i)=(t-i)/2$. Varying the first extension yields exactly $2^{\mathbf c}$ distinct lattice positive cones on $\mathbb{C}$, where $\mathbf c=|\mathbb{C}|$. The count concerns cones as subsets of $\mathbb{C}$.

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arXiv2026

Spectrum in alternative topological algebras and a new look at old theorems

In this paper, we consider real and complex algebras as well as algebras over general fields. In Section 2, we revisit and prove several results on (quadratic) algebras over general fields. As an example, we demonstrate that a quadratic algebra over a field of characteristic not $2$ is flexible if and only if it is proper--a concept introduced in this paper. In Section 3, we show how to develop the spectral theory in the context of complex (resp. real) one-sided alternative topological algebras. Along the way, spectral theory is used to prove several topological counterparts of the well-known theorems of Frobenius, Hurwitz, Gelfand-Mazur, and Zorn. This is done, for example, in the context of left (resp. right) alternative topological algebras whose duals separate their elements. In Section 4, we consider real and complex algebras in various topological settings and reconsider and prove several results. For instance, it is shown that given a $ 1 < k \in \mathbb{N}$, on any locally complex algebra, there exists a unique nonzero vector space norm, say, $\|.\|$, satisfying the identity $\|a^k\| = \|a\|^k$ on the algebra. In Section 5, among other things, we revisit and slightly strengthen the celebrated theorems of Frobenius, Zorn, Gelfand-Mazur, and Hurwitz, and also give slight extensions of their topological counterparts, e.g., theorems of Albert, Kaplansky, and Urbanik-Wright to name a few, in several

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