Related papers: Semiring identities in the semigroup $B_0$

An inverse semigroup $S$ is a semigroup in which every element has a unique inverse in the sense of semigroup theory, that is, if $a \in S$ then there exists a unique $b\in S$ such that $a = aba$ and $b = bab$. We say that an inverse…

We establish two sufficient conditions for an additively idempotent semiring to be nonfinitely based. As applications, we prove that two specific $4$-element additively idempotent semirings, $S_{(4,545)}$ and $S_{(4,634)}$, whose additive…

We say that a subring $R_0$ of a ring $R$ is semi-invariant if $R_0$ is the ring of invariants in $R$ under some set of ring endomorphisms of some ring containing $R$. We show that $R_0$ is semi-invariant if and only if there is a ring…

In this paper, we investigate semirings whose elements are either units or zero-divisors (nilpotents) with many examples. While comparing these semirings with their counterparts in ring theory, we observe that their behavior is different in…

For a groupoid $S$ with elements $a$ and $b$, if $ba = a$, then $b$ is a left identity of $a$ and $a$ is a right zero of $b$. We define the left identity set of $a$ to be the set of all left identities of $a$ in $S$, and similarly for the…

We show that the additively idempotent semiring $S_7^0$ has no finite basis for its equational theory. This answers an open problem posed by Jackson et al. (J. Algebra 611 (2022), 211--245).

We study the finite basis problem for 4-element additively idempotent semirings whose additive reducts are semilattices of height 1. Up to isomorphism, there are 58 such algebras. We show that 49 of them are finitely based and the remaining…

A semiring generalises the notion of a ring, replacing the additive abelian group structure with that of a commutative monoid. In this paper, we study a notion positioned between a ring and a semiring -- a semiring whose additive monoid is…

We exhibit a 6-element semigroup that has no finite identity basis but nevertheless generates a variety whose finite membership problem admits a polynomial algorithm.

We consider semigroup algorithmic problems in finitely generated metabelian groups. Our paper focuses on three decision problems introduced by Choffrut and Karhum\"{a}ki (2005): the Identity Problem (does a semigroup contain a neutral…

In this paper, we investigate zero-divisor, nilpotent, idempotent, unit, small, and irreducible elements in semiring extensions such as amount, content, and monoid semialgebras. We also introduce new concepts such as the prime avoidance…

For every semilattice $\mathcal{S}=(S,+)$, the set $\mathrm{End}(\mathcal{S})$ of its endomorphisms forms a semiring under point-wise addition and composition. We prove that the semiring of all endomorphisms of the 3-element chain has no…

It is proved that the additive group of every semidistributive nearring $R$ with an identity is abelian and if R has no elements of order $2$, then the nearring $R$ actually is an associative ring.

Semifields are semirings in which every nonzero element has a multiplicative inverse. A rough classification uses the characteristic of the semifield, that is the isomorphism type of the semifield generated by the two neutral elements. For…

In this paper we characterize the existance of an ultra central approximate identity for $\ell^{1}(S)$, where $S$ is a uniformly locally finite inverse semigroup. As an application, for the Brandt semigroup $S=M^{0}(G,I)$ over a non-empty…

We establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. Applying this condition, we prove that the six-element additively idempotent semiring $SR_6$ has no finite basis for its identity.…

We prove that a semiring multiplicatively generated by its idempotents is commutative and Boolean, if every idempotent in the semiring has an orthogonal complement. We prove that a semiring additively generated by its idempotents is…

We present a new proof for the main claim made in the author's paper "On the identity bases of Brandt semigroups" (Ural. Gos. Univ. Mat. Zap. 14, no.1 (1985), 38--42); this claim provides an identity basis for an arbitrary Brandt semigroup…

In this paper, we describe finite, additively commutative, congruence simple semirings. The main result is that the only such semirings are those of order 2, zero-multiplication rings of prime order, matrix rings over finite fields, those…

For a commutative semiring S, by an S-algebra we mean a commutative semiring A equipped with a homomorphism from S to A. We show that the subvariety of S-algebras determined by the identities 1+2x=1 and x^2=x is closed under non-empty…