Problem: In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.
What is the inverse of every element called?
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Answer: unanswerable


Problem: Sleep does not affect acquisition or recall while one is awake. Therefore, sleep has the greatest effect on memory consolidation. During sleep, the neural connections in the brain are strengthened. This enhances the brain’s abilities to stabilize and retain memories. There have been several studies which show that sleep improves the retention of memory, as memories are enhanced through active consolidation. System consolidation takes place during slow-wave sleep (SWS). This process implicates that memories are reactivated during sleep, but that the process doesn’t enhance every memory. It also implicates that qualitative changes are made to the memories when they are transferred to long-term store during sleep. When you are sleeping, the hippocampus replays the events of the day for the neocortex. The neocortex then reviews and processes memories, which moves them into long-term memory. When you do not get enough sleep it makes it more difficult to learn as these neural connections are not as strong, resulting in a lower retention rate of memories. Sleep deprivation makes it harder to focus, resulting in inefficient learning. Furthermore, some studies have shown that sleep deprivation can lead to false memories as the memories are not properly transferred to long-term memory. Therefore, it is important to get the proper amount of sleep so that memory can function at the highest level. One of the primary functions of sleep is thought to be the improvement of the consolidation of information, as several studies have demonstrated that memory depends on getting sufficient sleep between training and test. Additionally, data obtained from neuroimaging studies have shown activation patterns in the sleeping brain that mirror those recorded during the learning of tasks from the previous day, suggesting that new memories may be solidified through such rehearsal.
Does sleep hurt a persons memory?
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Answer: There have been several studies which show that sleep improves the retention of memory,


Problem: The similarities between Czech and Slovak led to the languages being considered a single language by a group of 19th-century scholars who called themselves "Czechoslavs" (Čechoslováci), believing that the peoples were connected in a way which excluded German Bohemians and (to a lesser extent) Hungarians and other Slavs. During the First Czechoslovak Republic (1918–1938), although "Czechoslovak" was designated as the republic's official language both Czech and Slovak written standards were used. Standard written Slovak was partially modeled on literary Czech, and Czech was preferred for some official functions in the Slovak half of the republic. Czech influence on Slovak was protested by Slovak scholars, and when Slovakia broke off from Czechoslovakia in 1938 as the Slovak State (which then aligned with Nazi Germany in World War II) literary Slovak was deliberately distanced from Czech. When the Axis powers lost the war and Czechoslovakia reformed, Slovak developed somewhat on its own (with Czech influence); during the Prague Spring of 1968, Slovak gained independence from (and equality with) Czech. Since then, "Czechoslovak" refers to improvised pidgins of the languages which have arisen from the decrease in mutual intelligibility.
What was standard written Slovak partially modeled on?
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Answer:
literary Czech