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[[File:Semiorder.svg|thumb|An example of a semiorder, shown by its [[Hasse diagram]]. The horizontal blue lines indicate the spacing of the ''y''-coordinates of the points; two points are comparable when their ''y'' coordinates differ by at least one unit.]]
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The multi-blade knives supply quite a lot of completely [http://www.crkt.com/pocketknives custom knife] different blades, every serving a unique function, permitting for higher flexibility of use. You can have a set of serrated, non-serrated, blades together with a blunt letter-opener, all packaged in the same multi-blade knife. Multi-blade knives are best outfitted for use around the house, whereas additionally allowing for some primary outside use.<br><br>This category describes a knife that you just intend to have on your individual full time. This can be a knife it's important to lug round with you, so attributes like weight play a larger function in your determination. Moreover, dimension plays a role from both a consolation standpoint in addition to a [http://Csw1.vaniercollege.qc.ca/users/SwapMe/wiki/index.php?title=Sog_Automatic_Knives_Reviews legality standpoint]. A sturdy locking mechanism is essential because it would probably receive substantial long term use. With an everyday carry knife, you want equal parts performance, reliability and comfort. Go away the limitless struggling and grumbling over tightly sealed bins to others. Slit it open in a matter of moments together with your trusty pocket knife.<br><br>In the knife group Chris Reeve is synonymous with quality and innovation and no discussion of best pocket knives is complete without mentioning the Sebenza. The Chris Reeve Sebenza has lengthy been regarded by the business as one of the best folding knives cash can purchase. It has a titanium body lock design that mixes simplicity with durability and a blade produced from S35VN stainless steel that eternally retains its edge. 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The quality is really second to none and it’ll shave the hairs in your chin proper out of the field!<br><br>Positive if we were all filthy wealthy our knife buying experience can be an entire lot simpler. The reality for many of us is that we're limited to a finances and wish to get the utmost performance within these limits. Because of this we pay explicit consideration to worth for cash. Many knives available on the market right now are merely overpriced for what they are. The good news is that there are also loads of glorious value for money options and should you look exhausting sufficient it’s attainable to seek out various top of the range knives with out breaking the bank. We appreciate your comments<br><br>The Kershaw Leek is four″ long closed, has a three″ blade (actually, 2 7/8″ by my measurement), and is simply shy of seven″ long when opened. It's 1″ huge at its widest level, 3/eight″ thick at its thinnest point, and half″ thick at its thickest point (on the pocket clip). It weighs a modest 2.four oz. This knife disappears in my entrance pocket. It is quite a bit thinner than most knives I am used to, and I have huge hands, so that will take some adjusting to. The blade protrusion seen just above the clip offers a substitute for the thumb studs for opening.<br><br>We consider it our mission that can assist you discover the appropriate knife that meets your needs. First off, you should try our BIG interactive chart of pocket knives which tables all the most well-liked knives and their specs to help you evaluate, distinction and make an informed alternative. As well as you’ll find more detailed evaluations [http://www.thebestpocketknifereviews.com/sog-knives-review/ The Best Pocket Knife Reviews.com] of specific knives in our critiques part and within other articles. The knife weighs four.zero ounces, which is value it for all of the extra instruments you get. 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In [[order theory]], a branch of mathematics, a '''semiorder''' is a type of ordering that may be determined for a set of items with numerical scores by declaring two items to be incomparable when their scores are within a given [[margin of error]] of each other, and by using the numerical comparison of their scores when those scores are sufficiently far apart. Semiorders were introduced and applied in [[mathematical psychology]] by {{harvtxt|Luce|1956}} as a model of human preference without the assumption that indifference is [[transitive relation|transitive]]. They generalize [[strict weak ordering]]s, form a special case of [[partial order]]s and [[interval order]]s, and can be characterized among the partial orders by two forbidden four-item suborders.
 
==Definition==
Let ''X'' be a set of items, and let < be a [[binary relation]] on ''X''.
Items ''x'' and ''y'' are said to be ''incomparable'', written here as ''x''&nbsp;~&nbsp;''y'', if neither ''x''&nbsp;<&nbsp;''y'' nor ''y''&nbsp;<&nbsp;''x'' is true. Then the pair (''X'',<) is a semiorder if it satisfies the following three axioms:<ref>{{harvtxt|Luce|1956}} describes an equivalent set of four axioms, the first two of which combine the definition of incomparability and the first axiom listed here.</ref>
*For all ''x'' and ''y'', it is not possible for both ''x'' < ''y'' and ''y'' < ''x'' to be true. That is, < must be an [[irreflexive relation|irreflexive]], [[antisymmetric relation]]
*For all ''x'', ''y'', ''z'', and ''w'', if it is true that ''x''&nbsp;<&nbsp;''y'', ''y''&nbsp;~&nbsp;''z'', and ''z''&nbsp;<&nbsp;''w'', then it must also be true that ''x''&nbsp;<&nbsp;''w''.
*For all ''x'', ''y'', ''z'', and ''w'', if it is true that ''x''&nbsp;<&nbsp;''y'', ''y''&nbsp;<&nbsp;''z'', and ''y''&nbsp;~&nbsp;''w'', then it cannot also be true that ''x''&nbsp;~&nbsp;''w'' and ''z''&nbsp;~&nbsp;''w'' simultaneously.
It follows from the first axiom that ''x''&nbsp;~&nbsp;''x'', and therefore the second axiom (with ''y''&nbsp;=&nbsp;''z'') implies that < is a [[transitive relation]].
 
One may define a [[partial order]] (''X'',≤) from a semiorder (''X'',<) by declaring that {{nowrap|''x'' ≤ ''y''}} whenever either {{nowrap|''x'' < ''y''}} or {{nowrap|1=''x'' = ''y''}}. Of the axioms that a partial order is required to obey, reflexivity follows automatically from this definition, antisymmetry follows from the first semiorder axiom, and transitivity follows from the second semiorder axiom. Conversely, from a partial order defined in this way, the semiorder may be recovered by declaring that {{nowrap|''x'' < ''y''}} whenever {{nowrap|''x'' ≤ ''y''}} and {{nowrap|''x'' ≠ ''y''}}. The first of the semiorder axioms listed above follows automatically from the axioms defining a partial order, but the others do not. The second and third semiorder axioms forbid partial orders of four items forming two disjoint chains: the second axiom forbids two chains of two items each, while the third item forbids a three-item chain with one unrelated item.
 
==Utility==
The original motivation for introducing semiorders was to model human preferences without assuming (as strict weak orderings do) that incomparability is a [[transitive relation]]. For instance, if ''x'', ''y'', and ''z'' represent three quantities of the same material, and ''x'' and ''z'' differ by the smallest amount that is perceptible as a difference, while ''y'' is halfway between the two of them, then it is reasonable for a preference to exist between ''x'' and ''z'' but not between the other two pairs, violating transitivity.<ref>{{harvtxt|Luce|1956}}, p. 179.</ref>
 
Thus, suppose that ''X'' is a set of items, and ''u'' is a [[utility function]] that maps the members of ''X'' to [[real number]]s. A strict weak ordering can be defined on ''x'' by declaring two items to be incomparable when they have equal utilities, and otherwise using the numerical comparison, but this necessarily leads to a transitive incomparability relation. Instead, if one sets a numerical threshold (which may be normalized to 1) such that utilities within that threshold of each other are declared incomparable, then a semiorder arises.
 
Specifically, define a binary relation < from ''X'' and ''u'' by setting ''x'' < ''y'' whenever ''u''(''x'')&nbsp;≤&nbsp;''u''(''y'')&nbsp;&minus;&nbsp;1. Then (''X'',<) is a semiorder.<ref>{{harvtxt|Luce|1956}}, Theorem 3 describes a more general situation in which the threshold for comparability between two utilities is a function of the utility rather than being identically 1.</ref> It may equivalently be defined as the [[interval order]] defined by the intervals [''u''(''x''),''u''(''x'')&nbsp;+&nbsp;1].<ref>{{harvtxt|Fishburn|1970}}.</ref>
 
The converse is not necessarily true: for instance, if a semiorder (''X'',<) includes an [[uncountable]] [[total order|totally ordered subset]] then there do not exist sufficiently many sufficiently well-spaced real-numbers to represent this subset numerically. However, every finite semiorder can be defined from a utility function in this way.<ref>This result is typically credited to {{harvtxt|Scott|Suppes|1958}}; see, e.g., {{harvtxt|Rabinovitch|1977}}. However, {{harvtxt|Luce|1956}}, Theorem 2 proves a more general statement, that a finite semiorder can be defined from a utility function and a threshold function whenever a certain underlying weak order can be defined numerically. For finite semiorders, it is trivial that the weak order can be defined numerically with a unit threshold function.</ref> {{harvtxt|Fishburn|1973}} supplies a precise characterization of the semiorders that may be defined numerically.
 
==Other results==
The number of distinct semiorders on ''n'' unlabeled items is given by the [[Catalan number]]s
:<math>\frac{1}{n+1}\binom{2n}{n},</math><ref>{{harvtxt|Kim|Roush|1978}}.</ref>
while the number of semiorders on ''n'' labeled items is given by the sequence
:1, 1, 3, 19, 183, 2371, 38703, 763099, 17648823, ... {{OEIS|A006531}}.<ref>{{harvtxt|Chandon|Lemaire|Pouget|1978}}.</ref>
 
Any finite semiorder has [[order dimension]] at most three.<ref>{{harvtxt|Rabinovitch|1978}}.</ref>
 
Among all partial orders with a fixed number of elements and a fixed number of comparable pairs, the partial orders that have the largest number of [[linear extension]]s are semiorders.<ref>{{harvtxt|Fishburn|Trotter|1992}}.</ref>
 
Semiorders are known to obey the [[1/3–2/3 conjecture]]: in any finite semiorder that is not a total order, there exists a pair of elements ''x'' and ''y'' such that ''x'' appears earlier than ''y'' in between 1/3 and 2/3 of the linear extensions of the semiorder.<ref>{{harvtxt|Brightwell|1989}}.</ref>
 
The set of semiorders on an ''n''-element set is ''well-graded'': if two semiorders on the same set differ from each other by the addition or removal of ''k'' order relations, then it is possible to find a path of ''k'' steps from the first semiorder to the second one, in such a way that each step of the path adds or removes a single order relation and each intermediate state in the path is itself a semiorder.<ref>{{harvtxt|Doignon|Falmagne|1997}}.</ref>
 
==Notes==
{{reflist|colwidth=30em}}
 
==References==
*{{citation
| last = Brightwell | first = Graham R. | authorlink = Graham Brightwell
| doi = 10.1007/BF00353656
| issue = 4
| journal = [[Order (journal)|Order]]
| pages = 369–380
| title = Semiorders and the 1/3–2/3 conjecture
| volume = 5
| year = 1989}}.
*{{citation
| last1 = Chandon | first1 = J.-L.
| last2 = Lemaire | first2 = J.
| last3 = Pouget | first3 = J.
| mr = 517680
| issue = 62
| journal = Centre de Mathématique Sociale. École Pratique des Hautes Études. Mathématiques et Sciences Humaines
| pages = 61–80, 83
| title = Dénombrement des quasi-ordres sur un ensemble fini
| year = 1978}}.
*{{citation
| last1 = Doignon | first1 = Jean-Paul
| last2 = Falmagne | first2 = Jean-Claude | author2-link = Jean-Claude Falmagne
| doi = 10.1016/S0012-365X(96)00095-7
| mr = 1468838
| issue = 1-3
| journal = Discrete Mathematics
| pages = 35–44
| title = Well-graded families of relations
| volume = 173
| year = 1997}}.
*{{citation
| last = Fishburn | first = Peter C. | authorlink = Peter C. Fishburn
| mr = 0253942
| journal = J. Mathematical Psychology
| pages = 144–149
| title = Intransitive indifference with unequal indifference intervals
| volume = 7
| year = 1970}}.
*{{citation
| last = Fishburn | first = Peter C. | authorlink = Peter C. Fishburn
| mr = 0316322
| journal = J. Mathematical Psychology
| pages = 91–105
| title = Interval representations for interval orders and semiorders
| volume = 10
| year = 1973}}.
*{{citation
| last1 = Fishburn | first1 = Peter C. | author1-link = Peter C. Fishburn
| last2 = Trotter | first2 = W. T.
| doi = 10.1016/0012-365X(92)90036-F
| mr = 1171114
| issue = 1
| journal = Discrete Mathematics
| pages = 25–40
| title = Linear extensions of semiorders: a maximization problem
| volume = 103
| year = 1992}}.
*{{citation
| last1 = Kim | first1 = K. H.
| last2 = Roush | first2 = F. W.
| mr = 538212
| issue = 2
| journal = Journal of Combinatorics, Information &System Sciences
| pages = 58–61
| title = Enumeration of isomorphism classes of semiorders
| volume = 3
| year = 1978}}.
*{{citation
| last = Luce | first = R. Duncan | authorlink = R. Duncan Luce
| mr = 0078632
| journal = Econometrica
| pages = 178–191
| title = Semiorders and a theory of utility discrimination
| jstor = 1905751
| volume = 24
| year = 1956}}.
*{{citation
| last = Rabinovitch | first = Issie
| mr = 0437404
| issue = 2
| journal = J. Mathematical Psychology
| pages = 209–212
| title = The Scott-Suppes theorem on semiorders
| volume = 15
| year = 1977}}.
*{{citation
| last = Rabinovitch | first = Issie
| mr = 0498294
| issue = 1
| journal = Journal of Combinatorial Theory. Series A
| pages = 50–61
| title = The dimension of semiorders
| volume = 25
| year = 1978}}.
*{{citation
| last1 = Scott | first1 = Dana | author1-link = Dana Scott
| last2 = Suppes | first2 = Patrick | author2-link = Patrick Suppes
| mr = 0115919
| journal = The Journal of Symbolic Logic
| pages = 113–128
| title = Foundational aspects of theories of measurement
| volume = 23
| year = 1958}}.
 
==Additional reading==
*{{citation
| last1 = Pirlot | first1 = M.
| last2 = Vincke | first2 = Ph.
| mr = 1472236
| isbn = 0-7923-4617-3
| location = Dordrecht
| publisher = Kluwer Academic Publishers Group
| series = Theory and Decision Library. Series B: Mathematical and Statistical Methods
| title = Semiorders: Properties, representations, applications
| volume = 36
| year = 1997}}.
 
[[Category:Order theory]]
[[Category:Mathematical relations]]

Latest revision as of 20:43, 6 January 2015



The large hole within the handle serves the same objective as a thumb stud, to permit for one handed opening. The physique is a extremely textured nylon that's quite stiff. The rear of the knife has a lanyard gap. On the latest fashions they have improved the pocket clip to make it ambidextrous, so you can mount it to both facet of the knife. They've also upgraded the blade to a barely fancier stainless steel. There have been a number of small changes through the years, as they hold making small tweaks to the identical basic design.

The first thing it is best to contemplate is the variety of blades you need. There are a lot of options for each single-blade and multi-blade pocket knives. The one-blade knives come geared up with a spring-loaded mechanism. This makes positive the blade might be popped open at a finger’s press. The multi-blade knives supply quite a lot of completely custom knife different blades, every serving a unique function, permitting for higher flexibility of use. You can have a set of serrated, non-serrated, blades together with a blunt letter-opener, all packaged in the same multi-blade knife. Multi-blade knives are best outfitted for use around the house, whereas additionally allowing for some primary outside use.

This category describes a knife that you just intend to have on your individual full time. This can be a knife it's important to lug round with you, so attributes like weight play a larger function in your determination. Moreover, dimension plays a role from both a consolation standpoint in addition to a legality standpoint. A sturdy locking mechanism is essential because it would probably receive substantial long term use. With an everyday carry knife, you want equal parts performance, reliability and comfort. Go away the limitless struggling and grumbling over tightly sealed bins to others. Slit it open in a matter of moments together with your trusty pocket knife.

In the knife group Chris Reeve is synonymous with quality and innovation and no discussion of best pocket knives is complete without mentioning the Sebenza. The Chris Reeve Sebenza has lengthy been regarded by the business as one of the best folding knives cash can purchase. It has a titanium body lock design that mixes simplicity with durability and a blade produced from S35VN stainless steel that eternally retains its edge. As soon as opened and locked the knife feels as strong as a hard and fast blade and holds comfortably within the hand. The quality is really second to none and it’ll shave the hairs in your chin proper out of the field!

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The Kershaw Leek is four″ long closed, has a three″ blade (actually, 2 7/8″ by my measurement), and is simply shy of seven″ long when opened. It's 1″ huge at its widest level, 3/eight″ thick at its thinnest point, and half″ thick at its thickest point (on the pocket clip). It weighs a modest 2.four oz. This knife disappears in my entrance pocket. It is quite a bit thinner than most knives I am used to, and I have huge hands, so that will take some adjusting to. The blade protrusion seen just above the clip offers a substitute for the thumb studs for opening.

We consider it our mission that can assist you discover the appropriate knife that meets your needs. First off, you should try our BIG interactive chart of pocket knives which tables all the most well-liked knives and their specs to help you evaluate, distinction and make an informed alternative. As well as you’ll find more detailed evaluations The Best Pocket Knife Reviews.com of specific knives in our critiques part and within other articles. The knife weighs four.zero ounces, which is value it for all of the extra instruments you get. It makes an incredible on daily basis carry and basic use knife. I wouldn’t suggest this as a standalone survival knife, for similar reasons to the Leek.