The delta set of a numerical semigroup $S$, denoted $\Delta (S)$, is a factorization invariant that measures the complexity of the sets of lengths of elements in~$S$. We study the following problem: Which finite sets occur as the delta set of a numerical semigroup $S$? It is known that $\min \Delta (S) = \gcd \Delta (S)$ is a necessary condition. For any two-element set $\{d,td\}$ we produce a semigroup~$S$ with this delta set. We then show that, for $t\ge 2$, the set $\{d,td\}$ occurs as the delta set of some numerical semigroup of embedding dimension~3 if and only if $t=2$.

Let $A$ be a commutative Noetherian ring of dimension~$n$ and $P$ a projective $A$-module of rank~$n$ with trivial determinant. In \cite {BR1}, Bhatwadekar and Sridharan defined the $n$th Euler class group of~$A$ and studied the obstruction to the existence of unimodular element in~$P$. For $R=A[T]$ and $R=A[T,T^{-1}]$, the $n$th Euler class groups of $R$ are defined by Das and Keshari in \cite {mkd1, MK1}, under certain assumption on $A$ in the latter case. We define the $n$th Euler class group of the ring $R=A[T,1/f(T)]$, where $f(T)\in A[T]$ is a monic polynomial and the height of the Jacobson radical of $A$ is $\geq 2$. Also, we prove results similar to those in \cite {MK1}.

Boij-S\"oderberg theory has had a dramatic impact on commutative algebra. We determine explicit Boij-S\"oderberg coefficients for ideals with linear resolutions and illustrate how these arise from the usual Eliahou-Kervaire computations for Borel ideals. In addition, we explore a new numerical decomposition for resolutions based on a row-by-row approach; here, the coefficients of the Betti diagrams are not necessarily positive. Finally, we demonstrate how the Boij-S\"oderberg decomposition of an arbitrary homogeneous ideal with a pure resolution changes when multiplying the ideal by a homogeneous polynomial.

A lattice-ordered group (an $\ell $-group) $G(\oplus , \vee , \wedge )$ can naturally be viewed as a semiring $G(\vee ,\oplus )$. We give a full classification of (abelian) $\ell $-groups which are finitely generated as semirings by first showing that each such $\ell $-group has an order-unit so that we can use the results of Busaniche, Cabrer and Mundici~\cite {BCM}. Then, we carefully analyze their construction in our setting to obtain the classification in terms of certain $\ell $-groups associated to rooted trees (Theorem \ref {classify}).

This classification result has a number of interesting applications; for example, it implies a classification of finitely generated ideal-simple (commutative) semirings $S(+, \cdot )$ with idempotent addition and provides important information concerning the structure of general finitely generated ideal-simple (commutative) semirings, useful in obtaining further progress towards Conjecture~\ref {main-conj} discussed in \cite {BHJK, JKK}.

In this paper, we show that the polyomino ideal of a simple polyomino coincides with the toric ideal of a weakly chordal bipartite graph, and hence, it has a quadratic Gr\"obner basis with respect to a suitable monomial order.

Using the connections among almost complete intersection schemes, arithmetically Gorenstein schemes and schemes that are a union of complete intersections, we give a structure theorem for the arithmetically Cohen-Macaulay union of two complete intersections of codimension~2, of type $(d_1,e_1)$ and $(d_2,e_2)$ such that $\min \{d_1,e_1\}\ne \min \{d_2,e_2\}$. We apply the results for computing Hilbert functions and graded Betti numbers for such schemes.