# Distributive lattices

1. Jan 16, 2012

### jessicat

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

2. Jan 16, 2012

### micromass

Staff Emeritus
Suppose that $a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c)$ holds. Now, let's look at

$$(x\vee y)\wedge (x\vee z)$$

Apply the distributivity with $a=x\vee y,~b=x,~c=z$.

3. Jan 17, 2012

### jessicat

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...............

4. Jan 17, 2012

### micromass

Staff Emeritus
Isn't that pretty obvious?? It doesn't need a distributive lattice as well, it's true in any lattice.

If $A=\{a_1,...,a_n\}$ is a non-empty finite set, then

$$\bigvee A=a_1\vee ... \vee a_n$$

5. Jan 17, 2012

### jessicat

yesssssssssss it is!!!! :tongue:thank u again.....

6. Jan 23, 2012

### jessicat

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............ :)