- #1
jessicat
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i have a question regarding maths,I have an exercise ...let L be a lattice and we know that it is distributive i.e we know tha aΛ(bVc)=(aΛb)V(αΛc) how can we prove that aV(bΛc)=(aVb)Λ(αVc);;;;;; thanks
Maths: Prove Distributive Lattices
- Thread starter jessicat
- Start date
In summary, if you have a behavior automaton for a semiring of natural numbers and an alphabet with two letters, and you know the transitions from one state to another with weights 1 and 1, then the automaton is equal to 2.
- #2
- 22,183
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Suppose that [itex]a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c)[/itex] holds. Now, let's look at
[tex](x\vee y)\wedge (x\vee z)[/tex]
Apply the distributivity with [itex]a=x\vee y,~b=x,~c=z[/itex].
[tex](x\vee y)\wedge (x\vee z)[/tex]
Apply the distributivity with [itex]a=x\vee y,~b=x,~c=z[/itex].
- #3
jessicat
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thank you very much for the instructions , they helped me to solve the exercise 
...i have also read sth that I understand intituitively but i cannot prove formally : L is a lattice and A is a subset of L and we denote with VA and ΛA the supremum and the infinum whenever they exist.Then how can I prove the proposition ΄If L is distributive then VA and ΛA exist in L for every finite A subset of L...
...i have also read sth that I understand intituitively but i cannot prove formally : L is a lattice and A is a subset of L and we denote with VA and ΛA the supremum and the infinum whenever they exist.Then how can I prove the proposition ΄If L is distributive then VA and ΛA exist in L for every finite A subset of L...- #4
- 22,183
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jessicat said:thank you very much for the instructions , they helped me to solve the exercise
Isn't that pretty obvious?? It doesn't need a distributive lattice as well, it's true in any lattice.
If [itex]A=\{a_1,...,a_n\}[/itex] is a non-empty finite set, then
[tex]\bigvee A=a_1\vee ... \vee a_n[/tex]
- #5
jessicat
- 7
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yesssssssssss it is! thank u again...
thank u again...- #6
jessicat
- 7
- 0
hallo...now i have a problem to solve regarding weigthed automata-in semirings(the automata are defined by matrices).. so i have to find the behaviour automaton through the solution of linear system..for simplicity i have constructed the follwing weigthed automaton : in the semiring of natural numbers , for an alphabet with two letters A= (a,b) I took 2 states qo and q1, qo is the initial state with initial weight 1 and q1 is the final state with final weight 1..i have the transitions from qo to q1 with weight 1 (letter a) from q1 to q1 (letters a and b)..i have proved through the definition of its behaviour that its equal to 2 but i cannot understand how to prove tis with linear systems ..i have tried to several times but my results are not equal to 2...my question is unfortunately very specific..i didnt know were to post it in the forum or where to ask for some lekp ..i hope you can give me some instructions...thanks in advance... :)
Related to Maths: Prove Distributive Lattices
1. What are distributive lattices in mathematics?
Distributive lattices are algebraic structures that represent a partially ordered set with additional operations called meet and join. These operations have the properties of distributivity, meaning that one operation distributes over the other.
2. How do you prove that a lattice is distributive?
To prove that a lattice is distributive, you must show that the join and meet operations satisfy the laws of distributivity. This means that for any three elements of the lattice, x, y, and z, the following equations must hold: x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) and x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z).
3. What is an example of a distributive lattice?
An example of a distributive lattice is the lattice of subsets of a set, where the join operation is given by set union and the meet operation is given by set intersection. This lattice satisfies the laws of distributivity because the union operation distributes over the intersection operation.
4. Are all lattices distributive?
No, not all lattices are distributive. There are lattices that do not satisfy the laws of distributivity, such as the lattice of divisors of a number, where the join operation is given by the greatest common divisor and the meet operation is given by the least common multiple.
5. What is the significance of distributive lattices in mathematics?
Distributive lattices have many applications in mathematics, including in algebra, logic, and computer science. They are also important in abstract algebra and can be used to define other algebraic structures, such as Boolean algebras. In addition, distributive lattices have connections to other areas of mathematics, such as topology and combinatorics.
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