Total order

In mathematics, a total order, simple order, linear order, connex order, or full order is a binary relation on some set X, antisymmetric, a connex relation. A set paired with a total order is called a chain, a ordered set, a ordered set, a linearly ordered set, or a loset. Formally, a binary relation ≤ is a total order on a set X if the following statements hold for all a, b and c in X: Antisymmetry If a ≤ b and b ≤ a a = b. Antisymmetry eliminates uncertain cases when both a precedes b precedes a. A relation having the connex property means that any pair of elements in the set of the relation are comparable under the relation; this means that the set can be diagrammed as a line of elements, giving it the name linear. The connex property implies reflexivity, i.e. a ≤ a. Therefore, a total order is a partial order, as, for a partial order, the connex property is replaced by the weaker reflexivity property. An extension of a given partial order to a total order is called a linear extension of that partial order.

For each total order ≤ there is an associated asymmetric transitive semiconnex relation <, called a strict total order or strict semiconnex order, which can be defined in two equivalent ways: a < b if a ≤ b and a ≠ b a < b if not b ≤ a Properties: The relation is transitive: a < b and b < c implies a < c. The relation is trichotomous: one of a < b, b < a and a = b is true. The relation is a strict weak order. We can start by choosing < as a transitive trichotomous binary relation. We can define or explain the way a set is ordered by any of these four relations; the letters of the alphabet ordered by the standard dictionary order, e.g. A < B < C etc. Any subset of a ordered set X is ordered for the restriction of the order on X; the unique order on the empty set, ∅, is a total order. Any set of cardinal numbers or ordinal numbers. If X is any set and f an injective function from X to a ordered set f induces a total ordering on X by setting x1 < x2 if and only if f < f. The lexicographical order on the Cartesian product of a family of ordered sets, indexed by a well ordered set, is itself a total order.

The set of real numbers ordered by the usual "less than" or "greater than" relations is ordered, hence so are the subsets of natural numbers and rational numbers. Each of these can be shown to be the unique smallest example of a ordered set with a certain property,: The natural numbers comprise the smallest non-empty ordered set with no upper bound; the integers comprise the smallest non-empty ordered set with neither an upper nor a lower bound. The rational numbers comprise the smallest ordered set, dense in the real numbers; the definition of density used here says that for every a and b in the real numbers such that a < b, there is a q in the rational numbers such that a < q < b. The real numbers comprise the smallest unbounded ordered set, connected in the order topology. Ordered fields are ordered by definition, they include the real numbers. Every ordered field contains an ordered subfield, isomorphic to the rational numbers. Any Dedekind-complete ordered; the term chain is a synonym for a ordered set, but the term is used to mean a ordered subset of some ordered set, for example in Zorn's lemma.

An ascending chain is a ordered set having a minimal element, while a descending chain is a ordered set having a maximal element. Given a set S with a partial order ≤, an infinite descending chain is an infinite decreasing sequence of elements x1 > x2 >.... As an example, in the set of integers, the chain −1, −2, −3... is an infinite descending chain, but there exists no infinite descending chain on the natural numbers, as every chain of natural numbers has a minimal element. If a ordered set does not possess any infinite descending chains, it is said to satisfy the descending chain condition. Assuming the axiom of choice, the descending chain condition on a ordered set is equivalent to requiring that the corresponding strict order is well-founded. A stronger condition, th

29th Battalion (Australia)

The 29th Battalion was an infantry battalion of the Australian Army. First formed in 1915 for service during the First World War as part of the Australian Imperial Force, it fought in the trenches of the Western Front in France and Belgium before being disbanded in late 1918 to provide reinforcements for other depleted Australian units. In 1921, following the demobilisation of the AIF, the battalion was re-raised as a unit of Australia's part-time military forces, based in Melbourne, before being amalgamated with the 22nd Battalion in 1930, it was re-raised in its own right and, following the outbreak of the Second World War, undertook garrison duties in Australia before being amalgamated with the 46th Battalion to form the 29th/46th Battalion in late 1942, subsequently seeing service against the Japanese in New Guinea and on New Britain. The 29th Battalion was formed during the First World War, being raised in Victoria as part of the Australian Imperial Force on 10 August 1915. Under the command of Lieutenant Colonel Alfred Bennett, an officer with over 20 years service in the part-time military forces, the battalion undertook initial training at Seymour and later Broadmeadows Camp along with the three other battalions of the 8th Brigade, to which it was assigned.

In November 1915, the battalion embarked upon the troopship HMAT Ascanius in Port Melbourne and departed Australian waters, disembarking at Port Suez, Egypt on 7 December 1915. The battalion arrived in the Middle East too late to take part in the fighting at Gallipoli, as a result they were used to undertake defensive duties to protect the Suez Canal from Ottoman forces, they undertook a comprehensive training program and by the time their orders arrived to transfer to Europe in June 1916, they had reached their peak. They subsequently embarked the troopship HMT Tunisian in Alexandria. Upon the battalion's arrival in Egypt, the 8th Brigade had been unattached at divisional level, but in early 1916, it was assigned to the 5th Division, after a reorganisation that saw the AIF expanded from two infantry divisions to five; the battalion afterwards was transported by rail to Hazebrouck. On 8 July the 5th Division was called up to the front from training behind lines in order to replace the battalions of the Australian 4th Division which were being transferred to the Somme.

The 29th Battalion undertook a difficult two-day 29-mile approach march over cobbled roads with loads of up to 70–75 pounds before arriving at the front on the night of 10/11 July. Taking up a position between Boutillerie and Condonerrie in the Bois Grenier, they relieved the 13th Battalion and on 19 July subsequently took part in an attack against the German positions around the "Delangre Farm", being held by the 21st Bavarian Reserve Infantry Regiment. Following the attack, the battalion held the line for another 11 days, beating off a heavy German counterattack on 20 July, before they were relieved. During their introduction to trench warfare, the 29th Battalion lost 52 men killed in action, another 164 men wounded. For the next two and half years they fought in a number of major battles in the trenches along the Western Front including Polygon Wood and the St Quentin Canal, as well as playing a supporting role in a number of others including Bullecourt and Morlancourt. During the Allied Hundred Days Offensive, launched on 8 August 1918, the battalion took part in the 8th Brigade's advance up the treacherous Morcourt Valley, subsequently achieving a considerable feat by capturing the town of Vauvillers.

The battalion fought its last battle of the war in late September alongside the US 30th Infantry Division, when they breached the German defences along the Hindenburg Line as part of the final Allied offensive of the war. Aimed at the Le Catelet Line near Bellicourt, the battalion began its advance on Nauroy on 29 September, moving on the left flank beside elements of the US 117th Infantry Regiment, with the 32nd Battalion in support; the attack proved successful and 59 prisoners were captured along with four field guns and a quantity of German small arms. Against this the battalion lost 63 wounded. Following this, they were withdrawn from the front line. By this time casualties amongst the Australian Corps had reached critical level and as a result many battalions—from an authorised strength of over 1,000 men—were only able to field between 300 and 400; as a result, the decision was made to reduce the number of infantry battalions in each brigade from four to three by disbanding one battalion and using its personnel to reinforce the others.

The 29th Battalion was one of those chosen to be broken up and as a result on 19 October 1918, the 29th Battalion was disbanded. The majority of the battalion's personnel—29 officers and 517 other ranks—were transferred to the 32nd Battalion as reinforcements. During its service on the Western Front, the battalion suffered 485 men killed and another 1,399 men wounded. Members of the battalion received the following decorations: three Distinguished Service Orders and one Bar, one Member of the Order of the British Empire, 20 Military Crosses, 17 Distinguished Conduct Medals, 94 Military Medals with three Bars, three Meritorious Service Medals, 17 Mentions in Despatches and five awards from other Allied nations; the 29th Battalion was bestowed 19 battle honours in 1927 for its involvement in the war. The battalion was re-raised in 1921 as part the re-organisation of the Australian military that took place at that time, with the battalion becoming a part-time unit of the Citizen Forces, assigned to the 4th Brigade, 3rd Division.

Upon formation, the battalion drew its personnel from four previously

Chinatown Fair

Chinatown Fair Family Fun Center is a video arcade center located on Mott Street in Chinatown, Manhattan. The arcade catered toward competitive fighting games; the original arcade opened in 1944 and closed in February 2011, but reopened in May 2012 under different management. Chinatown Fair has been regarded as New York City's "last great arcade". Chinatown Fair opened in 1944, taking over the first floor of the popular Port Arthur Chinese Restaurant building located at 7-9 Mott Street in New York City. Chinatown Fair operated as a penny arcade and small museum for many years, before becoming a video game arcade in the 1970s. Indian immigrant Sam Palmer purchased the business in 1982 after having a "religious vision". One of its first attractions was a dancing chicken; the dancing chicken was replaced with a tic-tac-toe playing chicken, retired in the early 2000s. In the 1970s and 1980s, the arcade hosted many retro generation games including Pac-Man and Space Invaders. In 1991, after the release of Street Fighter II: The World Warrior, Chinatown Fair switched focus to competitive fighting games.

By the late 1990s Street Fighter, Marvel vs. Capcom, The King of Fighters, Soulcalibur and other fighting games were played. Top players such as Justin Wong, NYChrisG, Michael "Yipes" Mendoza, Sanford Kelly frequented the establishment; as of 2010, The New York Times wrote that Chinatown Fair was among the last video arcades in the city. Video arcades have been in decline with the rise of home video games; the newspaper reported that it became "a center for all the outcasts in the city to bond over their shared love" of classic arcade and fighting video games no longer popular in modern arcades, with titles including the original Street Fighter II, The King of Fighters, Ms. Pac-Man. Other groups congregated around Dance Dance Revolution machines and racing games. Unlike the norm at other arcades, where winning players continue until deposed, Chinatown Fair players play a maximum of three or four continuous games. In February 2011, Chinatown Fair closed down. On May 5, 2012 over a year it reopened under a new name "Chinatown Fair Family Fun Center" with new manager and part owner, Lonnie Sobel.

Former competitive players criticized the new arcade for catering toward casual players, with the new ownership explaining that they were targeting a new clientele. Competitive fighting game players relocated to Next Level, a Brooklyn arcade owned by Chinatown Fair's former manager, which opened in 2011; the arcade is the focus of the 2015 documentary, The Lost Arcade, released in New York City on August 12, 2016. Official website