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| [[File:Series-parallel partial order.svg|thumb|360px|A series-parallel partial order, shown as a [[Hasse diagram]].]]
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| In [[order theory|order-theoretic]] mathematics, a '''series-parallel partial order''' is a [[partially ordered set]] built up from smaller series-parallel partial orders by two simple composition operations.<ref name="bgr">{{citation
| |
| | last1 = Bechet | first1 = Denis
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| | last2 = De Groote | first2 = Philippe
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| | last3 = Retoré | first3 = Christian
| |
| | contribution = A complete axiomatisation for the inclusion of series-parallel partial orders
| |
| | doi = 10.1007/3-540-62950-5_74
| |
| | pages = 230–240
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| | publisher = Springer-Verlag
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| | series = Lecture Notes in Computer Science
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| | title = Rewriting Techniques and Applications
| |
| | volume = 1232
| |
| | year = 1997}}.</ref><ref name="m">{{citation
| |
| | last = Möhring | first = Rolf H.
| |
| | editor-last = Rival | editor-first = Ivan | editor-link = Ivan Rival
| |
| | isbn = 978-0-7923-0007-6
| |
| | title = Algorithms and Order: Proceedings of the NATO Advanced Study Institute on Algorithms and Order, Ottawa, Canada, May 31-June 13, 1987
| |
| | pages = 105–194
| |
| | publisher = Springer-Verlag
| |
| | series = NATO Science Series C
| |
| | contribution = Computationally tractable classes of ordered sets
| |
| | volume = 255
| |
| | year = 1989}}.</ref>
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| | |
| The series-parallel partial orders may be characterized as the N-free finite partial orders; they have [[order dimension]] at most two.<ref name="bgr"/><ref name="vtl">{{citation | |
| | last1 = Valdes | first1 = Jacobo
| |
| | last2 = Tarjan | first2 = Robert E. | author2-link = Robert Tarjan
| |
| | last3 = Lawler | first3 = Eugene L. | author3-link = Eugene Lawler
| |
| | doi = 10.1137/0211023
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| | issue = 2
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| | journal = [[SIAM Journal on Computing]]
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| | pages = 298–313
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| | title = The recognition of series parallel digraphs
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| | volume = 11
| |
| | year = 1982}}.</ref> They include [[weak order]]s and the [[reachability]] relationship in [[Tree (graph theory)|directed trees]] and directed [[series-parallel graph]]s.<ref name="m"/><ref name="vtl"/> The [[comparability graph]]s of series-parallel partial orders are [[cograph]]s.<ref name="m"/><ref name="j"/>
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| | |
| Series-parallel partial orders have been applied in [[machine learning]] of event sequencing in [[time series]] data,<ref name="mm"/> transmission sequencing of [[multimedia]] data,<ref name="accdc"/> and throughput maximization in [[dataflow programming]].<ref name="cnns"/>
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| Series-parallel partial orders have also been called multitrees;<ref name="j">{{citation
| |
| | last = Jung | first = H. A.
| |
| | title = On a class of posets and the corresponding comparability graphs
| |
| | journal = [[Journal of Combinatorial Theory]], Series B
| |
| | volume = 24
| |
| | year = 1978
| |
| | issue = 2
| |
| | pages = 125–133
| |
| | id = {{MathSciNet | id = 0491356}}
| |
| | doi = 10.1016/0095-8956(78)90013-8}}.</ref> however, that name is ambiguous: [[multitree]]s also refer to partial orders with no four-element diamond suborder<ref>{{citation
| |
| | last1 = Furnas | first1 = George W.
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| | last2 = Zacks | first2 = Jeff
| |
| | contribution = Multitrees: enriching and reusing hierarchical structure
| |
| | doi = 10.1145/191666.191778
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| | pages = 330–336
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| | title = Proc. SIGCHI conference on Human Factors in Computing Systems (CHI '94)
| |
| | year = 1994}}.</ref> and to other structures formed from multiple trees.
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| | |
| ==Definition==
| |
| Consider ''P'' and ''Q'', two partially ordered sets. The series composition of ''P'' and ''Q'', written {{nowrap|''P''; ''Q''}},<ref name="accdc"/> {{nowrap|''P'' * ''Q''}},<ref name="m"/> or {{nowrap|''P'' ⧀ ''Q''}},<ref name="bgr"/>is the partially ordered set whose elements are the [[disjoint union]] of the elements of ''P'' and ''Q''. In {{nowrap|''P''; ''Q''}}, two elements ''x'' and ''y'' that both belong to ''P'' or that both belong to ''Q'' have the same order relation that they do in ''P'' or ''Q'' respectively. However, for every pair ''x'', ''y'' where ''x'' belongs to ''P'' and ''y'' belongs to ''Q'', there is an additional order relation {{nowrap|''x'' ≤ ''y''}} in the series composition. Series composition is an [[associative operation]]: one can write {{nowrap|''P''; ''Q''; ''R''}} as the series composition of three orders, without ambiguity about how to combine them pairwise, because both of the parenthesizations {{nowrap|(''P''; ''Q''); ''R''}} and {{nowrap|''P''; (''Q''; ''R'')}} describe the same partial order. However, it is not a [[commutative operation]], because switching the roles of ''P'' and ''Q'' will produce a different partial order that reverses the order relations of pairs with one element in ''P'' and one in ''Q''.<ref name="bgr"/>
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| | |
| The parallel composition of ''P'' and ''Q'', written ''P'' || ''Q'',<ref name="accdc"/> {{nowrap|''P'' + ''Q''}},<ref name="m"/> or {{nowrap|''P'' ⊕ ''Q''}},<ref name="bgr"/> is defined similarly, from the disjoint union of the elements in ''P'' and the elements in ''Q'', with pairs of elements that both belong to ''P'' or both to ''Q'' having the same order as they do in ''P'' or ''Q'' respectively. In ''P'' || ''Q'', a pair ''x'', ''y'' is incomparable whenever ''x'' belongs to ''P'' and ''y'' belongs to ''Q''. Parallel composition is both commutative and associative.<ref name="bgr"/>
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| | |
| The class of series-parallel partial orders is the set of partial orders that can be built up from single-element partial orders using these two operations. Equivalently, it is the smallest set of partial orders that includes the single-element partial order and is [[Closure (mathematics)|closed]] under the series and parallel composition operations.<ref name="bgr"/><ref name="m"/>
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| | |
| A [[weak order]] is the series parallel partial order obtained from a sequence of composition operations in which all of the parallel compositions are performed first, and then the results of these compositions are combined using only series compositions.<ref name="m"/>
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| | |
| ==Forbidden suborder characterization==
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| The partial order ''N'' with the four elements ''a'', ''b'', ''c'', and ''d'' and the three order relations {{nowrap|''a'' ≤ ''b'' ≥ ''c'' ≤ ''d''}} is an example of a [[Fence (mathematics)|fence]] or zigzag poset; its [[Hasse diagram]] has the shape of the capital letter "N". It is not series-parallel, because there is no way of splitting it into the series or parallel composition of two smaller partial orders. A partial order ''P'' is said to be N-free if there does not exist a set of four elements in ''P'' such that the restriction of ''P'' to those elements is order-isomorphic to ''N''. The series-parallel partial orders are exactly the nonempty finite N-free partial orders.<ref name="bgr"/><ref name="m"/><ref name="vtl"/>
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| | |
| It follows immediately from this (although it can also be proven directly) that any nonempty restriction of a series-parallel partial order is itself a series-parallel partial order.<ref name="bgr"/>
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| | |
| ==Order dimension==
| |
| The [[order dimension]] of a partial order ''P'' is the minimum size of a realizer of ''P'', a set of [[linear extension]]s of ''P'' with the property that, for every two distinct elements ''x'' and ''y'' of ''P'', {{nowrap|''x'' ≤ ''y''}} in ''P'' if and only if ''x'' has an earlier position than ''y'' in every linear extension of the realizer. Series-parallel partial orders have order dimension at most two. If ''P'' and ''Q'' have realizers {''L''<sub>1</sub>, ''L''<sub>2</sub>} and {''L''<sub>3</sub>, ''L''<sub>4</sub>}, respectively, then {''L''<sub>1</sub>''L''<sub>3</sub>, ''L''<sub>2</sub>''L''<sub>4</sub>} is a realizer of the series composition {{nowrap|''P''; ''Q''}}, and {''L''<sub>1</sub>''L''<sub>3</sub>, ''L''<sub>4</sub>''L''<sub>2</sub>} is a realizer of the parallel composition ''P'' || ''Q''.<ref name="m"/><ref name="vtl"/> A partial order is series-parallel if and only if it has a realizer in which one of the two permutations is the identity and the other is a [[separable permutation]].
| |
| | |
| It is known that a partial order ''P'' has order dimension two if and only if there exists a conjugate order ''Q'' on the same elements, with the property that any two distinct elements ''x'' and ''y'' are comparable on exactly one of these two orders. In the case of series parallel partial orders, a conjugate order that is itself series parallel may be obtained by performing a sequence of composition operations in the same order as the ones defining ''P'' on the same elements, but performing a series composition for each parallel composition in the decomposition of ''P'' and vice versa. More strongly, although a partial order may have many different conjugates, every conjugate of a series parallel partial order must itself be series parallel.<ref name="m"/>
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| | |
| ==Connections to graph theory==
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| Any partial order may be represented (usually in more than one way) by a [[directed acyclic graph]] in which there is a path from ''x'' to ''y'' whenever ''x'' and ''y'' are elements of the partial order with {{nowrap|''x'' ≤ ''y''}}. The graphs that represent series-parallel partial orders in this way have been called vertex series parallel graphs, and their [[transitive reduction]]s (the graphs of the [[covering relation]]s of the partial order) are called minimal vertex series parallel graphs.<ref name="vtl"/> Directed trees and (two-terminal) [[series parallel graph]]s are examples of minimal vertex series parallel graphs; therefore, series parallel partial orders may be used to represent reachability relations in directed trees and series parallel graphs.<ref name="m"/><ref name="vtl"/>
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| The [[comparability graph]] of a partial order is the [[undirected graph]] with a vertex for each element and an undirected edge for each pair of distinct elements ''x'', ''y'' with either {{nowrap|''x'' ≤ ''y''}} or {{nowrap|''y'' ≤ ''x''}}. That is, it is formed from a minimal vertex series parallel graph by forgetting the [[orientation (graph theory)|orientation]] of each edge. The comparability graph of a series-partial order is a [[cograph]]: the series and parallel composition operations of the partial order give rise to operations on the comparability graph that form the disjoint union of two subgraphs or that connect two subgraphs by all possible edges; these two operations are the basic operations from which cographs are defined. Conversely, every cograph is the comparability graph of a series-parallel partial order. If a partial order has a cograph as its comparability graph, then it must be a series-parallel partial order, because every other kind of partial order has an N suborder that would correspond to an induced four-vertex path in its comparability graph, and such paths are forbidden in cographs.<ref name="m"/><ref name="j"/>
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| | |
| ==Computational complexity==
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| It is possible to use the forbidden suborder characterization of series-parallel partial orders as a basis for an algorithm that tests whether a given binary relation is a series-parallel partial order, in an amount of time that is linear in the number of related pairs.<ref name="m"/><ref name="vtl"/> Alternatively, if a partial order is described as the [[reachability]] order of a [[directed acyclic graph]], it is possible to test whether it is a series-parallel partial order, and if so compute its transitive closure, in time proportional to the number of vertices and edges in the transitive closure; it remains open whether the time to recognizer series-parallel reachability orders can be improved to be linear in the size of the input graph.<ref>{{citation
| |
| | last1 = Ma | first1 = Tze-Heng
| |
| | last2 = Spinrad | first2 = Jeremy
| |
| | doi = 10.1007/BF00383402
| |
| | issue = 2
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| | journal = Order
| |
| | pages = 175–183
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| | title = Transitive closure for restricted classes of partial orders
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| | volume = 8
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| | year = 1991}}.</ref>
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| If a series-parallel partial order is represented as an [[expression tree]] describing the series and parallel composition operations that formed it, then the elements of the partial order may be represented by the leaves of the expression tree. A comparison between any two elements may be performed algorithmically by searching for the [[lowest common ancestor]] of the corresponding two leaves; if that ancestor is a parallel composition, the two elements are incomparable, and otherwise the order of the series composition operands determines the order of the elements. In this way, a series-parallel partial order on ''n'' elements may be represented in O(''n'') space with O(1) time to determine any comparison value.<ref name="m"/>
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| | |
| It is [[NP-complete]] to test, for two given series-parallel partial orders ''P'' and ''Q'', whether ''P'' contains a restriction isomorphic to ''Q''.<ref name="vtl"/>
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| | |
| Although the problem of counting the number of linear extensions of an arbitrary partial order is [[Sharp-P-complete|#P-complete]],<ref>{{citation
| |
| | last1 = Brightwell | first1 = Graham R.
| |
| | last2 = Winkler | first2 = Peter | author2-link = Peter Winkler
| |
| | doi = 10.1007/BF00383444
| |
| | issue = 3
| |
| | journal = [[Order (journal)|Order]]
| |
| | pages = 225–242
| |
| | title = Counting linear extensions
| |
| | volume = 8
| |
| | year = 1991}}.</ref> it may be solved in polynomial time for series-parallel partial orders. Specifically, if ''L''(''P'') denotes the number of linear extensions of a partial order ''P'', then ''L''(''P''; ''Q'') = ''L''(''P'')''L''(''Q'') and
| |
| :<math>L(P||Q)=\frac{(|P|+|Q|)!}{|P|!|Q|!} L(P)L(Q),</math>
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| so the number of linear extensions may be calculated using an expression tree with the same form as the decomposition tree of the given series-parallel order.<ref name="m"/>
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| | |
| ==Applications==
| |
| {{harvtxt|Mannila|Meek|2000}} use series-parallel partial orders as a model for the sequences of events in [[time series]] data. They describe [[machine learning]] algorithms for inferring models of this type, and demonstrate its effectiveness at inferring course prerequisites from student enrollment data and at modeling web browser usage patterns.<ref name="mm">{{citation
| |
| | last1 = Mannila | first1 = Heikki | author1-link = Heikki Mannila
| |
| | last2 = Meek | first2 = Christopher
| |
| | contribution = Global partial orders from sequential data
| |
| | doi = 10.1145/347090.347122
| |
| | pages = 161–168
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| | title = Proc. 6th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD 2000)
| |
| | year = 2000}}.</ref>
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|
| |
| {{harvtxt|Amer|Chassot|Connolly|Diaz|1994}} argue that series-parallel partial orders are a good fit for modeling the transmission sequencing requirements of [[multimedia]] presentations. They use the formula for computing the number of linear extensions of a series-parallel partial order as the basis for analyzing multimedia transmission algorithms.<ref name="accdc">{{citation
| |
| | last1 = Amer | first1 = Paul D.
| |
| | last2 = Chassot | first2 = Christophe
| |
| | last3 = Connolly | first3 = Thomas J.
| |
| | last4 = Diaz | first4 = Michel
| |
| | last5 = Conrad | first5 = Phillip
| |
| | doi = 10.1109/90.336326
| |
| | issue = 5
| |
| | journal = IEEE/ACM Transactions on Networking
| |
| | pages = 440–456
| |
| | title = Partial-order transport service for multimedia and other applications
| |
| | volume = 2
| |
| | year = 1994}}.</ref>
| |
| | |
| {{harvtxt|Choudhary|Narahari|Nicol|Simha|1994}} use series-parallel partial orders to model the task dependencies in a [[dataflow]] model of massive data processing for [[computer vision]]. They show that, by using series-parallel orders for this problem, it is possible to efficiently construct an optimized schedule that assigns different tasks to different processors of a [[parallel computing]] system in order to optimize the throughput of the system.<ref name="cnns">{{citation
| |
| | last1 = Choudhary | first1 = A. N.
| |
| | last2 = Narahari | first2 = B.
| |
| | last3 = Nicol | first3 = D. M.
| |
| | last4 = Simha | first4 = R.
| |
| | doi = 10.1109/71.273050
| |
| | issue = 4
| |
| | journal = IEEE Transactions on Parallel and Distributed Systems
| |
| | pages = 439–445
| |
| | title = Optimal processor assignment for a class of pipelined computations
| |
| | volume = 5
| |
| | year = 1994}}.</ref>
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| | |
| A class of orderings somewhat more general than series-parallel partial orders is provided by [[PQ tree]]s, data structures that have been applied in algorithms for testing whether a graph is [[planar graph|planar]] and recognizing [[interval graph]]s.<ref>{{citation
| |
| | last1 = Booth | first1 = Kellogg S.
| |
| | last2 = Lueker | first2 = George S.
| |
| | doi = 10.1016/S0022-0000(76)80045-1
| |
| | issue = 3
| |
| | journal = [[Journal of Computer and System Sciences]]
| |
| | pages = 335–379
| |
| | title = Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms
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| | volume = 13
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| | year = 1976}}.</ref> A ''P'' node of a PQ tree allows all possible orderings of its children, like a parallel composition of partial orders, while a ''Q'' node requires the children to occur in a fixed linear ordering, like a series composition of partial orders. However, unlike series-parallel partial orders, PQ trees allow the linear ordering of any ''Q'' node to be reversed.
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| ==References==
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| {{reflist}}
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| [[Category:Order theory]]
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| [[Category:Mathematical relations]]
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Kershaw Ken Onion Leek - The leek series is one of the hottest among the many model of Kershaw. The onion has a very glossy design made by Ken Onion. It encompasses a very good Velocity Safe system, which the proprietor can easily and simply deploy the blade with one hand. That is the perfect knife for everyday carry, it has a “reliable” clip which seems to be onerous to search out in most folding knives. It’s easy to say, you received’t find a folding knife at this worth with such a smooth design, good blade and total a strong-made knife. $36.29
A couple of minutes with an Arkansas whetstone and the carbon metal blade is sharp sufficient to cut paper. It got here with a fairly keen edge from the manufacturing unit and only somewhat end work was needed to get it scary sharp.The blade just isn't made out of chrome steel, however will be with an optionally available improve to one of many fancier models. Retaining a skinny coating of oil or wax on the blade will help forestall rust from being a problem. For lower than twelve bucks you do not get every part, but what you do get is sharp.
The primary actual knife I ever bought was the Kershaw Black Gulch Boy Scout model from the late 80s. It was loopy sharp out of the box and clearly a quality instrument to be handled with respect. That knife was in my pocket or pack for years, on many an adventure. Things have definitely changed. These days, Kershaw has a brand new excessive-end series of knives, the Zero Tolerance line. They are premium gadgets, with titanium and G-10 handles, in addition to ELMAX steel blades that run on a ball bearing meeting so they simply glide open.
Cardsharp two is a new improved superlight and tremendous sharp utility knife, dimension of a credit score card. Current capabilities encompass stiffer, polypropylene physique and unique primary safety lock. Only a few ingenious folding operation metamorphosis the card in an tasteful pocket utility useful resource. Slimmer and lighter than an frequent knife. The additional lengthy stainless metal surgical blade ensures for a longer interval lasting rust price-free sharpness. The superlight polypropylene physique accommodates a designed-in protective sheath that can assist shield towards injuries or blunting. Cardsharp2 is an ingenious dialogue piece. A pointy notion that slips safely and securely inside your wallet or bundle bag.
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The tear drop rear of the knife is ideal for nestling into your palm, sitting just below the muscle that controls your thumb. And not using a sharp or point edge, you get plenty of again up from the shape of your hand, especially in piercing cuts the place grip is extra necessary. In slicing and roll cuts the nice rounded bolster helps out as properly. When a knife pattern is this old, you have to figure that it stuck round for a motive. The good grip is one purpose why the Barlow remains to be round.
The primary electrical can opener Kitchen Collection sells is the "Hamilton Seaside Clean Touch Can Opener." All you must do with this electrical can opener is attach the lid of the can to the magnetic blade lever and press down on the lever while best type of throwing knives the can opener opens your can without leaving sharp edges. Then all you do is lift up on the lever pull the open can away from the electrical can opener and throw away the smoothed edged lid. This could opener is made out of chrome and plastic and weighs solely four pounds.