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| In [[logic]] and [[universal algebra]], '''Post's lattice''' denotes the [[lattice (order)|lattice]] of all [[clone (algebra)|clone]]s on a two-element set {0, 1}, ordered by [[inclusion (set theory)|inclusion]]. It is named for [[Emil Leon Post|Emil Post]], who published a complete description of the lattice in 1941.<ref>E. L. Post, ''The two-valued iterative systems of mathematical logic'', Annals of Mathematics studies, no. 5, Princeton University Press, Princeton 1941, 122 pp.</ref> The relative simplicity of Post's lattice is in stark contrast to the lattice of clones on a three-element (or larger) set, which has the [[cardinality of the continuum]], and a complicated inner structure. A modern exposition of Post's result can be found in Lau (2006).<ref>D. Lau, ''Function algebras on finite sets: Basic course on many-valued logic and clone theory'', Springer, New York, 2006, 668 pp. ISBN 978-3-540-36022-3</ref>
| | Today, there are several other types of web development and blogging software available to design and host your website blogs online and that too in minutes, if not hours. Thus, it is important to keep pace with this highly advanced age and have a regular interaction with your audience to keep a strong hold in the business market. If you have any questions concerning where and just how to make use of [http://www.jenligne.com/profile/makaminski wordpress backup plugin], you could contact us at the website. A pinch of tablet centric strategy can get your Word - Press site miles ahead of your competitors, so here are few strategies that will give your Wordpress websites and blogs an edge over your competitors:. If you're using Wordpress and want to make your blog a "dofollow" blog, meaning that links from your blog pass on the benefits of Google pagerank, you can install one of the many dofollow plugins available. This particular wordpress plugin is essential for not only having the capability where you improve your position, but to enhance your organic searches for your website. <br><br>Right starting from social media support to search engine optimization, such plugins are easily available within the Word - Press open source platform. If you wish to sell your services or products via internet using your website, you have to put together on the website the facility for trouble-free payment transfer between customers and the company. It allows Word - Press users to easily use HTML5 the element enable native video playback within the browser. Furthermore, with the launch of Windows 7 Phone is the smart phone market nascent App. This can be done by using a popular layout format and your unique Word - Press design can be achieved in other elements of the blog. <br><br>The entrepreneurs can easily captivate their readers by using these versatile themes. Note: at a first glance WP Mobile Pro themes do not appear to be glamorous or fancy. You've got invested a great cope of time developing and producing up the topic substance. Newer programs allow website owners and internet marketers to automatically and dynamically change words in their content to match the keywords entered by their web visitors in their search queries'a feat that they cannot easily achieve with older software. Search engine optimization pleasant picture and solution links suggest you will have a much better adjust at gaining considerable natural site visitors. <br><br>Word - Press installation is very easy and hassle free. php file in the Word - Press root folder and look for this line (line 73 in our example):. Next you'll go by way of to your simple Word - Press site. If you just want to share some picture and want to use it as a dairy, that you want to share with your friends and family members, then blogger would be an excellent choice. Word - Press offers constant updated services and products, that too, absolutely free of cost. <br><br>He loves sharing information regarding wordpress, Majento, Drupal and Joomla development tips & tricks. As a website owner, you can easily manage CMS-based website in a pretty easy and convenient style. Just download it from the website and start using the same. It is a fact that Smartphone using online customers do not waste much of their time in struggling with drop down menus. The 2010 voting took place from July 7 through August 31, 2010. |
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| ==Basic concepts==
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| A [[Boolean function]], or [[logical connective]], is an ''n''-ary [[operation (mathematics)|operation]] {{nowrap|''f'': '''2'''<sup>''n''</sup> → '''2'''}} for some {{nowrap|''n'' ≥ 1}}, where '''2''' denotes the two-element set {0, 1}. Particular Boolean functions are the [[projection (set theory)|projection]]s
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| :<math>\pi_k^n(x_1,\dots,x_n)=x_k,</math> | |
| and given an ''m''-ary function ''f'', and ''n''-ary functions ''g''<sub>1</sub>, ..., ''g''<sub>''m''</sub>, we can construct another ''n''-ary function
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| :<math>h(x_1,\dots,x_n)=f(g_1(x_1,\dots,x_n),\dots,g_m(x_1,\dots,x_n)),</math>
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| called their [[function composition|composition]]. A set of functions closed under composition, and containing all projections, is called a [[clone (algebra)|clone]].
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| Let ''B'' be a set of connectives. The functions which can be defined by a [[formula (mathematical logic)|formula]] using [[propositional variable]]s and connectives from ''B'' form a clone [''B''], indeed it is the smallest clone which includes ''B''. We call [''B''] the clone ''generated'' by ''B'', and say that ''B'' is the ''basis'' of [''B'']. For example, [¬, ⋀] are all Boolean functions, and [0, 1, ⋀, ⋁] are the monotone functions.
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| We use the operations ¬ ([[negation]]), ⋀ ([[logical conjunction|conjunction]] or [[meet (mathematics)|meet]]), ⋁ ([[disjunction]] or [[join (mathematics)|join]]), → ([[material conditional|implication]]), ↔ ([[logical biconditional|biconditional]]), + ([[exclusive disjunction]] or [[Boolean ring]] [[addition]]), ↛ ([[material nonimplication|nonimplication]]), ?: (the ternary [[?:|conditional operator]]) and the constant unary functions 0 and 1. Moreover, we need the [[threshold function]]s
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| :<math>\mathrm{th}^n_k(x_1,\dots,x_n)=\begin{cases}1&\text{if }\bigl|\{i\mid x_i=1\}\bigr|\ge k,\\
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| 0&\text{otherwise.}\end{cases}</math>
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| For example, th<sub>1</sub><sup>''n''</sup> is the large disjunction of all the variables ''x''<sub>''i''</sub>, and th<sub>''n''</sub><sup>''n''</sup> is the large conjunction. Of particular importance is the [[majority function]]
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| :<math>\mathrm{maj}=\mathrm{th}^3_2=(x\land y)\lor(x\land z)\lor(y\land z).</math>
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| We denote elements of '''2'''<sup>''n''</sup> (i.e., truth-assignments) as vectors: {{nowrap|1='''a''' = (''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)}}. The set '''2'''<sup>''n''</sup> carries a natural [[Cartesian product|product]] [[Boolean algebra (structure)|Boolean algebra]] structure. That is, ordering, meets, joins, and other operations on ''n''-ary truth assignments are defined pointwise:
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| :<math>(a_1,\dots,a_n)\le(b_1,\dots,b_n)\iff a_i\le b_i\text{ for every }i=1,\dots,n,</math>
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| :<math>(a_1,\dots,a_n)\land(b_1,\dots,b_n)=(a_1\land b_1,\dots,a_n\land b_n).</math>
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| ==Naming of clones==
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| [[intersection (set theory)|Intersection]] of an arbitrary number of clones is again a clone. It is convenient to denote intersection of clones by simple [[wiktionary:juxtaposition|juxtaposition]], i.e., the clone {{nowrap|''C''<sub>1</sub> ⋂ ''C''<sub>2</sub> ⋂ ... ⋂ ''C''<sub>''k''</sub>}} is denoted by ''C''<sub>1</sub>''C''<sub>2</sub>...''C''<sub>''k''</sub>. Some special clones are introduced below:
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| *M is the set of [[monotone function|monotone]] functions: {{nowrap|''f''('''a''') ≤ ''f''('''b''')}} for every {{nowrap|'''a''' ≤ '''b'''}}.
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| *D is the set of [[de Morgan dual|self-dual]] functions: {{nowrap|1=¬''f''('''a''') = ''f''(¬'''a''')}}.
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| *A is the set of [[affine transformation|affine]] functions: the functions satisfying
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| ::<math>f(a_1,\dots,a_{i-1},c,a_{i+1},\dots,a_n)=f(a_1,\dots,d,a_{i+1},\dots)\ \Rightarrow\ f(b_1,\dots,c,b_{i+1},\dots)=f(b_1,\dots,d,b_{i+1},\dots)</math>
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| :for every ''i'' ≤ ''n'', '''a''', '''b''' ∈ '''2'''<sup>''n''</sup>, and ''c'', ''d'' ∈ '''2'''. Equivalently, the functions expressible as {{nowrap|1=''f''(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) = ''a''<sub>0</sub> + ''a''<sub>1</sub>''x''<sub>1</sub> + ... + ''a''<sub>''n''</sub>''x''<sub>''n''</sub>}} for some ''a''<sub>0</sub>, '''a'''.
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| *U is the set of ''essentially unary'' functions, i.e., functions which depend on at most one input variable: there exists an ''i'' = 1, ..., ''n'' such that {{nowrap|1=''f''('''a''') = ''f''('''b''')}} whenever {{nowrap|1=''a''<sub>''i''</sub> = ''b''<sub>''i''</sub>}}.
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| *Λ is the set of ''conjunctive'' functions: {{nowrap|1=''f''('''a''' ⋀ '''b''') = ''f''('''a''') ⋀ ''f''('''b''')}}. The clone Λ consists of the conjunctions <math>f(x_1,\dots,x_n)=\bigwedge_{i\in I}x_i</math> for all subsets ''I'' of {1, ..., ''n''} (including the empty conjunction, i.e., the constant 1), and the constant 0.
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| *V is the set of ''disjunctive'' functions: {{nowrap|1=''f''('''a''' ⋁ '''b''') = ''f''('''a''') ⋁ ''f''('''b''')}}. Equivalently, V consists of the disjunctions <math>f(x_1,\dots,x_n)=\bigvee_{i\in I}x_i</math> for all subsets ''I'' of {1, ..., ''n''} (including the empty disjunction 0), and the constant 1.
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| *For any ''k'' ≥ 1, T<sub>0</sub><sup>''k''</sup> is the set of functions ''f'' such that
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| ::<math>\mathbf a^1\land\cdots\land\mathbf a^k=\mathbf 0\ \Rightarrow\ f(\mathbf a^1)\land\cdots\land f(\mathbf a^k)=0.</math>
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| :Moreover, <math>\mathrm{T}_0^\infty=\bigcap_{k=1}^\infty\mathrm{T}_0^k</math> is the set of functions bounded above by a variable: there exists ''i'' = 1, ..., ''n'' such that {{nowrap|''f''('''a''') ≤ ''a''<sub>''i''</sub>}} for all '''a'''.
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| :As a special case, {{nowrap|1=P<sub>0</sub> = T<sub>0</sub><sup>1</sup>}} is the set of ''0-preserving'' functions: {{nowrap|1=''f''('''0''') = 0}}.
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| *For any ''k'' ≥ 1, T<sub>1</sub><sup>''k''</sup> is the set of functions ''f'' such that
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| ::<math>\mathbf a^1\lor\cdots\lor\mathbf a^k=\mathbf 1\ \Rightarrow\ f(\mathbf a^1)\lor\cdots\lor f(\mathbf a^k)=1,</math>
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| :and <math>\mathrm{T}_1^\infty=\bigcap_{k=1}^\infty\mathrm{T}_1^k</math> is the set of functions bounded below by a variable: there exists ''i'' = 1, ..., ''n'' such that {{nowrap|''f''('''a''') ≥ ''a''<sub>''i''</sub>}} for all '''a'''.
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| :The special case {{nowrap|1=P<sub>1</sub> = T<sub>1</sub><sup>1</sup>}} consists of the ''1-preserving'' functions: {{nowrap|1=''f''('''1''') = 1}}.
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| *The largest clone of all functions is denoted ⊤, the smallest clone (which contains only projections) is denoted ⊥, and {{nowrap|1=P = P<sub>0</sub>P<sub>1</sub>}} is the clone of ''constant-preserving'' functions.
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| ==Lattice==
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| The set of all clones is a [[closure system]], hence it forms a [[complete lattice]]. The lattice is [[countably infinite]], and all its members are finitely generated. All the clones are listed in the table below.
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| [[Image:Post-lattice.svg|thumb|right|500px|[[Hasse diagram]] of Post's lattice]]
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| [[Image:Post-lattice-centre.svg|thumb|right|300px|Central part of the lattice]]
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| {| class="wikitable"
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| ! clone !! one of its bases
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| |-
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| |⊤||⋁, ¬
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| |-
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| |P<sub>0</sub>||⋁, +
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| |-
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| |P<sub>1</sub>||⋀, →
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| |-
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| |P||''x'' ? ''y'' : ''z''
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| |-
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| |T<sub>0</sub><sup>''k''</sup>, ''k'' ≥ 2||th<sub>''k''</sub><sup>''k''+1</sup>, ↛
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| |-
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| |T<sub>0</sub><sup>∞</sup>||↛
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| |-
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| |PT<sub>0</sub><sup>''k''</sup>, ''k'' ≥ 2||th<sub>''k''</sub><sup>''k''+1</sup>, ''x'' ⋀ (''y'' → ''z'')
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| |-
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| |PT<sub>0</sub><sup>∞</sup>||''x'' ⋀ (''y'' → ''z'')
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| |-
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| |T<sub>1</sub><sup>''k''</sup>, ''k'' ≥ 2||th<sub>2</sub><sup>''k''+1</sup>, →
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| |-
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| |T<sub>1</sub><sup>∞</sup>||→
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| |-
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| |PT<sub>1</sub><sup>''k''</sup>, ''k'' ≥ 2||th<sub>2</sub><sup>''k''+1</sup>, ''x'' ⋁ (''y'' + ''z'')
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| |-
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| |PT<sub>1</sub><sup>∞</sup>||''x'' ⋁ (''y'' + ''z'')
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| |-
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| |M||⋀, ⋁, 0, 1
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| |-
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| |MP<sub>0</sub>||⋀, ⋁, 0
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| |-
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| |MP<sub>1</sub>||⋀, ⋁, 1
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| |-
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| |MP||⋀, ⋁
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| |-
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| |MT<sub>0</sub><sup>''k''</sup>, ''k'' ≥ 2||th<sub>''k''</sub><sup>''k''+1</sup>, 0
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| |-
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| |MT<sub>0</sub><sup>∞</sup>||''x'' ⋀ (''y'' ⋁ ''z''), 0
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| |-
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| |MPT<sub>0</sub><sup>''k''</sup>, ''k'' ≥ 2||th<sub>''k''</sub><sup>''k''+1</sup> for ''k'' ≥ 3,<br />maj, ''x'' ⋀ (''y'' ⋁ ''z'') for ''k'' = 2
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| |-
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| |MPT<sub>0</sub><sup>∞</sup>||''x'' ⋀ (''y'' ⋁ ''z'')
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| |-
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| |MT<sub>1</sub><sup>''k''</sup>, ''k'' ≥ 2||th<sub>2</sub><sup>''k''+1</sup>, 1
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| |-
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| |MT<sub>1</sub><sup>∞</sup>||''x'' ⋁ (''y'' ⋀ ''z''), 1
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| |-
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| |MPT<sub>1</sub><sup>''k''</sup>, ''k'' ≥ 2||th<sub>2</sub><sup>''k''+1</sup> for ''k'' ≥ 3,<br />maj, ''x'' ⋁ (''y'' ⋀ ''z'') for ''k'' = 2
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| |-
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| |MPT<sub>1</sub><sup>∞</sup>||''x'' ⋁ (''y'' ⋀ ''z'')
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| |-
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| |Λ||⋀, 0, 1
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| |-
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| |ΛP<sub>0</sub>||⋀, 0
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| |-
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| |ΛP<sub>1</sub>||⋀, 1
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| |-
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| |ΛP||⋀
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| |-
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| |V||⋁, 0, 1
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| |-
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| |VP<sub>0</sub>||⋁, 0
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| |-
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| |VP<sub>1</sub>||⋁, 1
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| |-
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| |VP||⋁
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| |-
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| |D||maj, ¬
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| |-
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| |DP||maj, ''x'' + ''y'' + ''z''
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| |-
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| |DM||maj
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| |-
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| |A||↔, 0
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| |-
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| |AD||¬, ''x'' + ''y'' + ''z''
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| |-
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| |AP<sub>0</sub>||+
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| |-
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| |AP<sub>1</sub>||↔
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| |-
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| |AP||''x'' + ''y'' + ''z''
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| |-
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| |U||¬, 0
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| |-
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| |UD||¬
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| |-
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| |UM||0, 1
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| |-
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| |UP<sub>0</sub>||0
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| |-
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| |UP<sub>1</sub>||1
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| |-
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| |⊥||
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| |}
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| The eight infinite families have actually also members with ''k'' = 1, but these appear separately in the table: {{nowrap|1=T<sub>0</sub><sup>1</sup> = P<sub>0</sub>}}, {{nowrap|1=T<sub>1</sub><sup>1</sup> = P<sub>1</sub>}}, {{nowrap|1=PT<sub>0</sub><sup>1</sup> = PT<sub>1</sub><sup>1</sup> = P}}, {{nowrap|1=MT<sub>0</sub><sup>1</sup> = MP<sub>0</sub>}}, {{nowrap|1=MT<sub>1</sub><sup>1</sup> = MP<sub>1</sub>}}, {{nowrap|1=MPT<sub>0</sub><sup>1</sup> = MPT<sub>1</sub><sup>1</sup> = MP}}.
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| The lattice has a natural symmetry mapping each clone ''C'' to its dual clone {{nowrap|1=''C''<sup>''d''</sup> = {''f''<sup>d</sup> | ''f'' ∈ ''C''}}}, where {{nowrap|1=''f''<sup>''d''</sup>(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) = ¬''f''(¬''x''<sub>1</sub>, ..., ¬''x''<sub>''n''</sub>)}} is the [[de Morgan dual]] of a Boolean function ''f''. For example, {{nowrap|1=Λ<sup>''d''</sup> = V}}, {{nowrap|1=(T<sub>0</sub><sup>''k''</sup>)<sup>''d''</sup> = T<sub>1</sub><sup>''k''</sup>}}, and {{nowrap|1=M<sup>''d''</sup> = M}}.
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| ==Applications==
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| The complete classification of Boolean clones given by Post helps to resolve various questions about classes of Boolean functions. For example:
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| *An inspection of the lattice shows that the maximal clones different from ⊤ (often called '''Post's classes''') are M, D, A, P<sub>0</sub>, P<sub>1</sub>, and every proper subclone of ⊤ is contained in one of them. As a set ''B'' of connectives is [[functional completeness|functionally complete]] if and only if it generates ⊤, we obtain the following characterization: ''B'' is functionally complete iff it is not included in one of the five Post's classes.
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| *The [[satisfiability problem]] for Boolean formulas is [[NP-complete]] by [[Cook's theorem]]. Consider a restricted version of the problem: for a fixed finite set ''B'' of connectives, let ''B''-SAT be the algorithmic problem of checking whether a given ''B''-formula is satisfiable. Lewis<ref>[[Harry R. Lewis|H. R. Lewis]], ''Satisfiability problems for propositional calculi'', Mathematical Systems Theory 13 (1979), pp. 45–53.</ref> used the description of Post's lattice to show that ''B''-SAT is NP-complete if the function ↛ can be generated from ''B'' (i.e., {{nowrap|[''B''] ⊇ T<sub>0</sub><sup>∞</sup>}}), and in all the other cases ''B''-SAT is [[P (complexity)|polynomial-time]] decidable.
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| ==Variants==
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| Post originally did not work with the modern definition of clones, but with the so-called ''iterative systems'', which are sets of operations closed under substitution
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| :<math>h(x_1,\dots,x_{n+m-1})=f(x_1,\dots,x_{n-1},g(x_n,\dots,x_{n+m-1})),</math>
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| as well as permutation and identification of variables. The main difference is that iterative systems do not necessarily contain all projections. Every clone is an iterative system, and there are 20 non-empty iterative systems which are not clones. (Post also excluded the empty iterative system from the classification, hence his diagram has no least element and fails to be a lattice.) As another alternative, some authors work with the notion of a ''closed class'', which is an iterative system closed under introduction of dummy variables. There are four closed classes which are not clones: the empty set, the set of constant 0 functions, the set of constant 1 functions, and the set of all constant functions.
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| Composition alone does not allow to generate a nullary function from the corresponding unary constant function, this is the technical reason why nullary functions are excluded from clones in Post's classification. If we lift the restriction, we get more clones. Namely, each clone ''C'' in Post's lattice which contains at least one constant function corresponds to two clones under the less restrictive definition: ''C'', and ''C'' together with all nullary functions whose unary versions are in ''C''.
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| ==References==
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| <references />
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| [[Category:Universal algebra]]
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| [[Category:Logic]]
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Today, there are several other types of web development and blogging software available to design and host your website blogs online and that too in minutes, if not hours. Thus, it is important to keep pace with this highly advanced age and have a regular interaction with your audience to keep a strong hold in the business market. If you have any questions concerning where and just how to make use of wordpress backup plugin, you could contact us at the website. A pinch of tablet centric strategy can get your Word - Press site miles ahead of your competitors, so here are few strategies that will give your Wordpress websites and blogs an edge over your competitors:. If you're using Wordpress and want to make your blog a "dofollow" blog, meaning that links from your blog pass on the benefits of Google pagerank, you can install one of the many dofollow plugins available. This particular wordpress plugin is essential for not only having the capability where you improve your position, but to enhance your organic searches for your website.
Right starting from social media support to search engine optimization, such plugins are easily available within the Word - Press open source platform. If you wish to sell your services or products via internet using your website, you have to put together on the website the facility for trouble-free payment transfer between customers and the company. It allows Word - Press users to easily use HTML5 the element enable native video playback within the browser. Furthermore, with the launch of Windows 7 Phone is the smart phone market nascent App. This can be done by using a popular layout format and your unique Word - Press design can be achieved in other elements of the blog.
The entrepreneurs can easily captivate their readers by using these versatile themes. Note: at a first glance WP Mobile Pro themes do not appear to be glamorous or fancy. You've got invested a great cope of time developing and producing up the topic substance. Newer programs allow website owners and internet marketers to automatically and dynamically change words in their content to match the keywords entered by their web visitors in their search queries'a feat that they cannot easily achieve with older software. Search engine optimization pleasant picture and solution links suggest you will have a much better adjust at gaining considerable natural site visitors.
Word - Press installation is very easy and hassle free. php file in the Word - Press root folder and look for this line (line 73 in our example):. Next you'll go by way of to your simple Word - Press site. If you just want to share some picture and want to use it as a dairy, that you want to share with your friends and family members, then blogger would be an excellent choice. Word - Press offers constant updated services and products, that too, absolutely free of cost.
He loves sharing information regarding wordpress, Majento, Drupal and Joomla development tips & tricks. As a website owner, you can easily manage CMS-based website in a pretty easy and convenient style. Just download it from the website and start using the same. It is a fact that Smartphone using online customers do not waste much of their time in struggling with drop down menus. The 2010 voting took place from July 7 through August 31, 2010.