Maths: Prove Distributive Lattices

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Discussion Overview

The discussion revolves around proving properties of distributive lattices in mathematics, specifically focusing on exercises related to lattice operations and the existence of supremum and infimum for subsets. The thread also touches on a separate problem involving weighted automata in semirings.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant asks how to prove that if a lattice L is distributive, then the equation a∨(b∧c)=(a∨b)∧(a∨c) holds.
  • Another participant suggests applying distributivity to the expression (x∨y)∧(x∨z) using specific substitutions.
  • A participant expresses gratitude for instructions that helped solve the initial exercise and raises a question about proving that if L is distributive, then the supremum and infimum exist for every finite subset A of L.
  • Another participant challenges the necessity of the distributive property for the existence of supremum and infimum, arguing that it holds in any lattice.
  • A later reply confirms the previous participant's assertion, indicating agreement on the point raised.
  • A separate participant introduces a problem related to weighted automata in semirings, describing their construction and seeking help with proving results using linear systems.

Areas of Agreement / Disagreement

Participants generally agree on the existence of supremum and infimum in any lattice, but there is a lack of consensus on the necessity of the distributive property for this assertion. The discussion regarding the weighted automata problem remains unresolved, with the participant seeking further guidance.

Contextual Notes

Some assumptions regarding the definitions of lattice operations and the properties of semirings may not be explicitly stated, and the mathematical steps in proving the weighted automata behavior are not fully detailed.

Who May Find This Useful

Readers interested in lattice theory, mathematical proofs related to distributive properties, and those working on problems involving weighted automata in semirings may find this discussion relevant.

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
 
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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].
 
thank you very much for the instructions , they helped me to solve the exercise :smile:...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...
 
jessicat said:
thank you very much for the instructions , they helped me to solve the exercise :smile:...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...

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]
 
yesssssssssss it is! :-pthank u again...
 
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... :)
 

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