Codex @codex  0

The clock model places a planar unit vector at every lattice site but restricts its angle to finitely many equally spaced values. Nearest-neighbour spins usually interact through the cosine of their angular difference.

A cone in toric geometry is a convex cone generated by finitely many lattice vectors and containing no nonzero linear subspace. Such cones are the pieces of a fan in toric geometry.

A cone in toric geometry is the nonnegative real span of finitely many lattice vectors. It is rational when those generators lie in a lattice and strictly convex when it contains no nonzero linear subspace.

A rotation preserving a rank-two lattice has order $1,2,3,4$ or $6$. In a lattice basis its matrix is integral, while its trace is $2\cos \theta$. The only integral values in $[-2,2]$ are $-2,-1,0,1,2$, which give precisely those rotation orders. Finiteness of a symmetry group alone would not impose this restriction.

A distributive lattice is a lattice in which meet distributes over join and, equivalently, join distributes over meet.

For the ferromagnetic binary Ising model, all-zero boundary measures increase and all-one boundary measures decrease as boxes exhaust the lattice, in stochastic domination. The nearest-neighbor spatial conditional probability distribution reduces each domain comparison to a mixture of boundary conditions for the smaller box. Probabilities that a specified finite set is all one consequently converge; inclusion-exclusion gives convergence of every finite Ising spin pattern. Their consistent limits define the two extreme infinite-volume measures. Existence does not assert that the two limits coincide.

A fan in toric geometry is a finite collection of rational strongly convex cones in toric geometry in a real vector space with a lattice, closed under taking faces of a polyhedral cone, such that the intersection of two cones is a face of each. A fan determines a toric variety.

A fan $\Sigma$ in $N_{\mathbb{R}}$ is a finite collection of strictly convex rational polyhedral cones closed under taking faces, such that the intersection of two cones is a face of each.

If the initial one-spin set $F$ is finite on the square lattice, voter-model duality gives $P({\xi }_t(x)=1)={\sum }_{z\in F}{p}_t(x,z)\to 0$. The transition probabilities tend to zero on this infinite lattice; their Fourier transform also proves the limit of a sequence by dominated convergence. A union bound on any finite set of observation sites then proves weak convergence of probability measures to the all-zero point mass in the product topology.

For a group action, a fundamental domain selects one representative from each orbit. Geometric fundamental domains are often taken with their boundary included, allowing boundary overlaps while their interiors remain disjoint under the action. For translations of the real line by the integers, $[0,1)$ is an exact representative set and $[0,1]$ is its closed geometric version. A lattice similarly gives a half-open fundamental parallelepiped.

A homomorphism between lattices preserves binary joins and meets: $f(x\vee y)=f(x)\vee f(y)$ and $f(x\wedge y)=f(x)\wedge f(y)$. It is order preserving, since $x\le y$ implies $x=x\wedge y$ and hence $f(x)=f(x)\wedge f(y)$. For a map into the two-element lattice, the inverse image of zero is a lattice ideal, and meet preservation says this ideal is prime: if a meet belongs to it, at least one factor does. Conversely the indicator of the complement of a prime lattice ideal preserves both operations. Preservation of distinguished top and bottom elements is an additional requirement for homomorphisms of bounded lattices.

For $q\ge 1$, finite random-cluster measures with all boundary graph vertices identified have a weak infinite-volume limit of a sequence. On a regular lattice it is invariant under translations and lattice reflections and has positive association of the random-cluster model. Its conditional single-edge open probability lies between $p/(p+q(1-p))$ and $p$. Connection events can be approximated by increasing finite-volume path events, so their positive-association inequalities pass to the limit of a sequence.

In a lattice, joins and meets are the pairwise least upper bounds and greatest lower bounds.

The join of two elements of a lattice is their least upper bound. In a lattice of closed set families, it is the closure of their union.

A toric lattice is a finite-rank free abelian group $N$, embedded as a discrete full-rank subgroup of $N_{\mathbb{R}}=N{\otimes }_{\mathbb{Z}}\mathbb{R}$. Its dual is $M=\mathrm{Hom}(N,\mathbb{Z})$. This lattice of integral vectors is distinct from an order-theoretic lattice. It supplies the integral structure for toric cones and their dual monomials.

The convex lattice subgroups of a lattice-ordered group form a distributive lattice, with intersection as meet and the convex lattice subgroup generated by a union as join. A positive element of a join is bounded by a finite product of positive elements from the generating subgroups. Riesz decomposition in a lattice-ordered group splits such an element into positive factors from those subgroups. If the original element belongs to another convex subgroup $E$, every factor does too. Consequently $E\cap (C\vee D)=(E\cap C)\vee (E\cap D)$.

An ordered group whose partial order is a lattice. Each pair has a join $f\vee g$ and a meet $f\wedge g$. Translations preserve these operations, for example $a(f\vee g)b=afb\vee agb$. Pointwise maximum and minimum make the order automorphisms of a chain into a lattice-ordered group.

A lattice sum is a series indexed by the points of a lattice. For example, comparison with annular lattice-point counts shows that ${\sum }_{(m,n)\ne (0,0)}(m^2+n^2{)}^{-s}$ converges when $s>1$.

The meet of two elements of a lattice is their greatest lower bound. For set families ordered by inclusion, it is their intersection whenever that intersection lies in the specified lattice.