International Journal of Algebra and Computation

A regular semigroup is weakly generated by a set $X$ if it has no proper regular subsemigroup containing $X$. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup $\mathrm{\text{FI}}(X)$ weakly generated by $\left|X\right|$ idempotents such that all other regular semigroups weakly generated by $\left|X\right|$ idempotents are homomorphic images of $\mathrm{\text{FI}}(X)$. The semigroup $\mathrm{\text{FI}}(X)$ is defined by a presentation $〈\mathrm{\text{G}}(X),{\rho }_e\cup {\rho }_s〉$ and its structure is studied. Although each of the sets $\mathrm{\text{G}}(X)$, ${\rho }_e$, and ${\rho }_s$ is infinite for $\left|X\right|\ge 2$, we show that the word problem is decidable as each congruence class has a “canonical form”. If ${\mathrm{\text{FI}}}_n$ denotes $\mathrm{\text{FI}}(X)$ for $\left|X\right|=n$, we prove also that ${\mathrm{\text{FI}}}_2$ contains copies of all ${\mathrm{\text{FI}}}_n$ as subsemigroups. As a consequence, we conclude that (i) all regular semigroups weakly generated by a finite set of idempotents, which include all finitely idempotent generated regular semigroups, strongly divide ${\mathrm{\text{FI}}}_2$; and (ii) all finite semigroups divide ${\mathrm{\text{FI}}}_2$.

We discuss the isomorphism problem of projective schemes; given two projective schemes, can we algorithmically decide whether they are isomorphic? We give affirmative answers in the case of one-dimensional projective schemes, the case of smooth irreducible varieties with a big canonical sheaf or a big anti-canonical sheaf, and the case of K3 surfaces with a finite automorphism group. As related algorithmic problems, we also discuss decidability of positivity properties of invertible sheaves, and approximation of the nef cone and the pseudo-effective cone.

Let $D$ be a weighted oriented graph and $I(D)$ be its edge ideal. We provide one method to find all the minimal generators of $I_{\subseteq C}$, where $C$ is a maximal strong vertex cover of $D$ and $I_{\subseteq C}$ is the intersections of irreducible ideals associated to the strong vertex covers contained in $C$. If $D^{\prime }$ is an induced digraph of $D$, under a certain condition on the strong vertex covers of $D^{\prime }$ and $D$, we show that ${I(D^{\prime })}^{(s)}\ne {I(D^{\prime })}^s$ for some $s\ge 2$ implies ${I(D)}^{(s)}\ne {I(D)}^s$. We provide the necessary and sufficient condition for the equality of ordinary and symbolic powers of edge ideal of the union of two naturally oriented paths with a common sink vertex. We characterize all the maximal strong vertex covers of $D$ such that at most one edge is oriented into each of its vertices and $w(x)\ge 2$ if ${deg}_D(x)\ge 2$ for all $x\in V(D)$. Finally, if $D$ is a weighted rooted tree with the degree of root is $1$ and $w(x)\ge 2$ when ${deg}_D(x)\ge 2$ for all $x\in V(D)$, we show that ${I(D)}^{(s)}={I(D)}^s$ for all $s\ge 2$.

Epimorphisms in the category of partially ordered monoids may not be surjective. We prove that lax epimorphisms in the category of partially ordered monoids are surjective. We further show that lax epimorphisms in the said category coincide with coinserters. Our results also generalize to partially ordered semigroups.

In this paper the concept of local embeddability into finite structures (being LEF) is examined for the class of semigroups. The established results include the connections to the previously studied class of LEF groups, residual finiteness, linear semigroups and the preservation of being LEF under certain semigroup constructions, such as adjoining zero or identity and taking direct, semidirect and wreath products.

Let $S$ be a ring with a partial action $\alpha$ of a finite group $G$. We determine when a quotient group of $G$ gives rise to a partial action induced by $\alpha$ on a subring of $S$. As an application, we show that if $S/S^{\alpha }$ is an $\alpha$-partial Galois extension and $K$ is a normal subgroup of $G$, then under certain conditions $S^{{\alpha }_K}/S^{\alpha }$ under the partial action of $G/K$ induced by $\alpha$, denoted by ${\alpha }_{G/K}$, is a partial Galois extension. This result was just shown recently by Bagio et al., applying globalization to derive a partial action of $G/K$ on $S^{{\alpha }_K}$, totally different from the one presented in this paper arising from a Boolean ring generated by certain idempotents. We further show in either type of induced partial action of a quotient group that under certain conditions $S^{{\alpha }_K}/S^{\alpha }$ is a DeMeyer–Kanzaki ${\alpha }_{G/K}$-partial Galois extension whenever $S/S^{\alpha }$ is an $\alpha$-partial Galois Azumaya extension. A structure theorem for a partial Galois extension is also presented, namely, every partial Galois extension can be decomposed as a direct sum of Galois extensions and possibly a trivial partial Galois extension of type II.

We consider the action of the group ${\mathrm{\text{PGL}}}_4(K)$ on the smooth cubic surfaces of ${\mathbb{P}}_K^3$ ($K$ an algebraically closed field of characteristic zero). We classify, in an explicit way, all the smooth cubic surfaces with nontrivial stabilizer, the corresponding stabilizers and obtain a geometric description of each group in terms of permutations of the Eckardt points, of the $27$ lines or of the $45$ tritangent planes.

Let $M$ be a graph manifold containing a single JSJ torus $T$ and whose JSJ blocks are of the form $\mathrm{\Sigma }\times S^1$, where $\mathrm{\Sigma }$ is a compact orientable surface with boundary. We show that if $M$ does not admit a Riemannian metric of everywhere nonpositive sectional curvature, then there is an essential curve on $T$ such that any finite-dimensional linear representation of ${\pi }_1(M)$ maps an element representing that curve to a matrix all of whose eigenvalues are roots of $1$. In particular, this shows that ${\pi }_1(M)$ does not admit a faithful finite-dimensional unitary representation, and gives a new proof that ${\pi }_1(M)$ is not linear over any field of positive characteristic.

Let $G$ be a finite group and $p$ a fixed prime divisor of $\left|G\right|$. We prove that if every maximal subgroup of $G$ is nilpotent, or normal, or has $p^{\prime }$-order, then (1) $G$ is solvable; (2) $G$ has a Sylow tower; (3) there exists at most one prime divisor $q\ne p$ of $\left|G\right|$ such that $G$ is neither $q$-nilpotent nor $q$-closed.