| 31. | Every complete lattice is also a bounded lattice, which is to say that it has a greatest and least element.
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| 32. | The main property to derive this uniqueness is the following : For every in, is the least element of such that.
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| 33. | Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements.
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| 34. | One can define half-closed and closed intervals,, and by adjoining as a least element and / or as a greatest element.
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| 35. | By using e " instead of d " in the above definition, one defines the least element of " S ".
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| 36. | It is easy to see that 0 is the least element with respect to this order : for all " a ".
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| 37. | Here's the general proof : suppose " X " is a nonempty set that you wish to prove has a least element.
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| 38. | An example is given by the above divisibility order |, where 1 is the least element since it divides all other numbers.
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| 39. | The greatest element in this fiber is the discrete topology on " X " while the least element is the indiscrete topology.
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| 40. | An important property of the natural numbers is that they are well-ordered : every non-empty set of natural numbers has a least element.
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