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Given a poset 
$P$ and a standard closure operator 
$\unicode[STIX]{x1D6E4}:{\wp}(P)\rightarrow {\wp}(P)$, we give a necessary and sufficient condition for the lattice of 
$\unicode[STIX]{x1D6E4}$-closed sets of 
${\wp}(P)$ to be a frame in terms of the recursive construction of the 
$\unicode[STIX]{x1D6E4}$-closure of sets. We use this condition to show that, given a set 
${\mathcal{U}}$ of distinguished joins from 
$P$, the lattice of 
${\mathcal{U}}$-ideals of 
$P$ fails to be a frame if and only if it fails to be 
$\unicode[STIX]{x1D70E}$-distributive, with 
$\unicode[STIX]{x1D70E}$ depending on the cardinalities of sets in 
${\mathcal{U}}$. From this we deduce that if a poset has the property that whenever 
$a\wedge (b\vee c)$ is defined for 
$a,b,c\in P$ it is necessarily equal to 
$(a\wedge b)\vee (a\wedge c)$, then it has an 
$(\unicode[STIX]{x1D714},3)$-representation.
The jaggedness of an order ideal 
$I$ in a poset 
$P$ is the number of maximal elements in 
$I$ plus the number of minimal elements of 
$P$ not in 
$I$. A probability distribution on the set of order ideals of 
$P$ is toggle-symmetric if for every 
$p\in P$, the probability that 
$p$ is maximal in 
$I$ equals the probability that 
$p$ is minimal not in 
$I$. In this paper, we prove a formula for the expected jaggedness of an order ideal of 
$P$ under any toggle-symmetric probability distribution when 
$P$ is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan–López–Pflueger–Teixidor [Trans. Amer. Math. Soc., forthcoming. 2015, arXiv:1506.00516], who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp–Roby, of the antichain cardinality statistic for order ideals in partially ordered sets.
To a pair 
$P$ and 
$Q$ of finite posets we attach the toric ring 
$K[P,Q]$ whose generators are in bijection to the isotone maps from 
$P$ to 
$Q$. This class of algebras, called isotonian, are natural generalizations of the so-called Hibi rings. We determine the Krull dimension of these algebras and for particular classes of posets 
$P$ and 
$Q$ we show that 
$K[P,Q]$ is normal and that their defining ideal admits a quadratic Gröbner basis.
We study a natural generalization of the noncrossing relation between pairs of elements in 
$[n]$ to 
$k$-tuples in 
$[n]$ that was first considered by Petersen et al. [J. Algebra324(5) (2010), 951–969]. We give an alternative approach to their result that the flag simplicial complex on 
$\binom{[n]}{k}$ induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product 
$[k]\times [n-k]$ of two chains (also called Gelfand–Tsetlin polytope), and that it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). We then observe that this already implies the existence of a flag simplicial polytope generalizing the dual associahedron, whose Stanley–Reisner ideal is an initial ideal of the Grassmann–Plücker ideal, while previous constructions of such a polytope did not guarantee flagness nor reduced to the dual associahedron for 
$k=2$. On our way we provide general results about order polytopes and their triangulations. We call the simplicial complex the noncrossing complex, and the polytope derived from it the dual Grassmann associahedron. We extend results of Petersen et al. [J. Algebra324(5) (2010), 951–969] showing that the noncrossing complex and the Grassmann associahedron naturally reflect the relations between Grassmannians with different parameters, in particular the isomorphism
$G_{k,n}\cong G_{n-k,n}$. Moreover, our approach allows us to show that the adjacency graph of the noncrossing complex admits a natural acyclic orientation that allows us to define a Grassmann–Tamari order on maximal noncrossing families. Finally, we look at the precise relation of the noncrossing complex and the weak separability complex of Leclerc and Zelevinsky [Amer. Math. Soc. Transl.181(2) (1998), 85–108]; see also Scott [J. Algebra290(1) (2005), 204–220] among others. We show that the weak separability complex is not only a subcomplex of the noncrossing complex as noted by Petersen et al. [J. Algebra324(5) (2010), 951–969] but actually its cyclically invariant part.
For a semigroup 
$S$, let 
${S}^{1} $ be the semigroup obtained from 
$S$ by adding a new symbol 1 as its identity if 
$S$ has no identity; otherwise let 
${S}^{1} = S$. Mitsch defined the natural partial order 
$\leqslant $ on a semigroup 
$S$ as follows: for 
$a, b\in S$, 
$a\leqslant b$ if and only if 
$a= xb= by$ and 
$a= ay$ for some 
$x, y\in {S}^{1} $. In this paper, we characterise the natural partial order on some transformation semigroups. In these partially ordered sets, we determine the compatibility of their elements, and find all minimal and maximal elements.
A subset F of an ordered set X is a fibre of X if F intersects every maximal antichain of X. We find a lower bound on the function ƒ (D), the minimum fibre size in the distributive lattice D, in terms of the size of D. In particular, we prove that there is a constant c such that 
In the process we show that minimum fibre size is a monotone property for a certain class of distributive lattices. This fact depends upon being able to split every maximal antichain of this class of distributive lattices into two parts so that the lattice is the union of the upset of one part and the downset of the other.
