Lattice 2

[Rocki Eibl]

Alattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor.

Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These lattice-like structures all admit order-theoretic as well as algebraic descriptions.

Lattices have some connections to the family of group-like algebraic structures. Because meet and join both commute and associate, a lattice can be viewed as consisting of two commutative semigroups having the same domain. For a bounded lattice, these semigroups are in fact commutative monoids. The absorption law is the only defining identity that is peculiar to lattice theory. A bounded lattice can also be thought of as a commutative rig without the distributive axiom.

By commutativity, associativity and idempotence one can think of join and meet as operations on non-empty finite sets, rather than on pairs of elements. In a bounded lattice the join and meet of the empty set can also be defined (as 0 \displaystyle 0 and 1 , \displaystyle 1, respectively). This makes bounded lattices somewhat more natural than general lattices, and many authors require all lattices to be bounded.

In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function preserving binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set.

Any homomorphism of lattices is necessarily monotone with respect to the associated ordering relation; see Limit preserving function. The converse is not true: monotonicity by no means implies the required preservation of meets and joins (see Pic. 9), although an order-preserving bijection is a homomorphism if its inverse is also order-preserving.

Given the standard definition of isomorphisms as invertible morphisms, a lattice isomorphism is just a bijective lattice homomorphism. Similarly, a lattice endomorphism is a lattice homomorphism from a lattice to itself, and a lattice automorphism is a bijective lattice endomorphism. Lattices and their homomorphisms form a category.

A poset is called a complete lattice if all its subsets have both a join and a meet. In particular, every complete lattice is a bounded lattice. While bounded lattice homomorphisms in general preserve only finite joins and meets, complete lattice homomorphisms are required to preserve arbitrary joins and meets.

Every poset that is a complete semilattice is also a complete lattice. Related to this result is the interesting phenomenon that there are various competing notions of homomorphism for this class of posets, depending on whether they are seen as complete lattices, complete join-semilattices, complete meet-semilattices, or as join-complete or meet-complete lattices.

A conditionally complete lattice is a lattice in which every nonempty subset that has an upper bound has a join (that is, a least upper bound). Such lattices provide the most direct generalization of the completeness axiom of the real numbers. A conditionally complete lattice is either a complete lattice, or a complete lattice without its maximum element 1 , \displaystyle 1, its minimum element 0 , \displaystyle 0, or both.[4][5]

A lattice that satisfies the first or, equivalently (as it turns out), the second axiom, is called a distributive lattice.The only non-distributive lattices with fewer than 6 elements are called M3 and N5;[6] they are shown in Pictures 10 and 11, respectively. A lattice is distributive if and only if it does not have a sublattice isomorphic to M3 or N5.[7] Each distributive lattice is isomorphic to a lattice of sets (with union and intersection as join and meet, respectively).[8]

For an overview of stronger notions of distributivity that are appropriate for complete lattices and that are used to define more special classes of lattices such as frames and completely distributive lattices, see distributivity in order theory.

A finite lattice is modular if and only if it is both upper and lower semimodular. For a graded lattice, (upper) semimodularity is equivalent to the following condition on the rank function r : \displaystyle r\colon

In domain theory, it is natural to seek to approximate the elements in a partial order by "much simpler" elements. This leads to the class of continuous posets, consisting of posets where every element can be obtained as the supremum of a directed set of elements that are way-below the element. If one can additionally restrict these to the compact elements of a poset for obtaining these directed sets, then the poset is even algebraic. Both concepts can be applied to lattices as follows:

Both of these classes have interesting properties. For example, continuous lattices can be characterized as algebraic structures (with infinitary operations) satisfying certain identities. While such a characterization is not known for algebraic lattices, they can be described "syntactically" via Scott information systems.

Any set X \displaystyle X may be used to generate the free semilattice F X . \displaystyle FX. The free semilattice is defined to consist of all of the finite subsets of X , \displaystyle X, with the semilattice operation given by ordinary set union. The free semilattice has the universal property. For the free lattice over a set X , \displaystyle X, Whitman gave a construction based on polynomials over X \displaystyle X 's members.[10][11]

We now define some order-theoretic notions of importance to lattice theory. In the following, let x \displaystyle x be an element of some lattice L . \displaystyle L. x \displaystyle x is called:

The notions of ideals and the dual notion of filters refer to particular kinds of subsets of a partially ordered set, and are therefore important for lattice theory. Details can be found in the respective entries.

Definitely something that needs to be looked into. I am trying to see if I can sync Asana into Jira and then Jira into lattice simply because Jira and lattice sync but we cant sync from Asana to Lattice.

Yep, need this! Lattice is used on HR side and more like a whiteboard. But for our team to actually project manage our goals and collaborate we need to work in Lattice. Big pain to copy over each new goal to Asana, since we already have them in Lattice. Would be great to have collab across. Thanks! Thanks to Yas for getting me in forum!

The lattice add-on package is an implementation of Trellis graphics for R. It is a powerful and elegant high-level data visualization system with an emphasis on multivariate data. It is designed to meet most typical graphics needs with minimal tuning, but can also be easily extended to handle most nonstandard requirements.

I created gyroid and diamond structure in Creo by Engineering -> lattice -> formula driven. Then I exported this to STL file and printed it... But now I need to do some analysys in Ansys... so I need igs format (because Ansys do not support STL to do analyses). Is the way how to convert structure from STL (or prt, where I creted this structure) to iges format ? Or how to solve this problem ? How to do analyses of minimal surfaces strucutre if I can not create solid model from them ?

I already tried this... But my stl format have about 4 000 000 polygons and when I want to create igs this way Freecad still crashed. I have 16GB memory ram and I think it is not enought...Is there other solution ?

1_ Use CREO Simulation Live. It understands voxel-based geometry and you can do Structural, Modal, Thermal, and CFD analyses. In the CFD you can compute velocities, pressure distributions, and pressure drop but you can not yet conjugate heat transfer. It is amazing fast. Use the highest fidelity level and include internal bodies on the setup.

2 Export the STL and import it in ANSYS Discovery. You can set up the boundary conditions using named selection in design & explore modes. You can go to Refine mode if you want to go to ANSYS Mechanical Workbench or Fluent.

So, I "apply" the deformation by selecting the model and deleting the history. Now I export it to unity, and it looks correct. HOWEVER, the history deleting has also messed up the rigging - the model no longer has a "SkinnedMeshRenderer" attached, just a "MeshRenderer".

So you should be able to shift your lattice deformation underneath the skin, then export as FBX and the lattice deformation will be in the base mesh. I tested this with a simple example and it worked for me.

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