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Semigroup action

of a semigroup $S$ on a set $X$

A map from $X \times S \rightarrow X$, written $(x,s)$ or $x^s$ satisfying $$ (x,st) = ((x,s),t)\ . $$ For given $s$, the map $\rho_s : x \mapsto (x,s)$ is a transformation of $X$. The map $s \mapsto \rho_s$ is a semigroup morphism $\rho : S \rightarrow T_X$ where $T_X$ is the transformation semi-group on $X$: conversely, every such homomorphism gives rise to an action $(x,s) \mapsto (x)\rho_s$. If the homomorphism $\rho$ is injective the action is faithful: $S$ may be regarded as a subsemigroup of $T_X$.

If $S$ is a monoid with identity element $1_S$, the action is unital if in addition $$ (x,1_S) = x $$ for all $x \in X$. In this case the map$s \mapsto \rho_s$ is a monoid morphism.

Polygon (over a monoid)

$R$-polygon, operand

A non-empty set with a monoid of operators. More precisely, a non-empty set $A$ is called a left polygon over a monoid $R$ if for any $\lambda \in R$ and $a \in A$ the product $\lambda a \in A$ is defined, such that $$ \lambda (\mu a) = (\lambda\mu) a $$ and $$ 1a = a $$ for any $\lambda, \mu \in R$, $a \in A$. A right polygon is defined similarly. Specifying an $R$-polygon $A$ is equivalent to specifying a homomorphism $\phi$ from the monoid $R$ into the monoid of mappings of the set $A$ into itself that transforms 1 to the identity mapping. Here $\lambda a = b$ if and only if $$ \phi(\lambda)a = b \ . $$ In particular, each non-empty set may be considered as a polygon over the monoid of its mappings into itself. Therefore, polygons are closely related to the representation of semi-groups by transformations: cf. Transformation semi-group.

If $A$ is a universal algebra whose signature $\Omega$ contains only unary operations, then $A$ can be converted into a polygon over the free monoid $F$ generated by $\Omega$ by putting $$ f_1 \cdots f_n a = f_1(\cdots(f_n(a)\cdots) $$

for any $f_i \in \Omega$, $a \in A$. If $\Omega$ is the set of input signals for an automaton having set of states $A$, then $A$ is similarly transformed into an $F$-polygon (cf. Automata, algebraic theory of).

A mapping $\phi$ of an $R$-polygon $A$ into an $R$-polygon $B$ is called a homomorphism if $\phi(\lambda a) = \lambda \phi(a)$ for any $\lambda \in R$ and $a \in A$. For $A=B$ one arrives at the definition of an endomorphism. All endomorphisms of a polygon $A$ form a monoid, and $A$ can be considered as a polygon over it.

An equivalence relation $\theta$ on an $R$-polygon $A$ is called a congruence if $(a,b) \in \theta$ implies $\lambda a,\lambda b) \in \theta$ for any $\lambda \in R$. The set of congruence classes of $\theta$ is naturally transformed into an $R$-polygon, called a quotient polygon of the polygon $A$ and denoted by $A/\theta$. If $A$ is a polygon over $R$, then in $R$ one can define a relation $\mathop{Ann} A$ by putting $(\lambda,\mu) \in \mathop{Ann} A$ if $\lambda a = \mu a$ for all $a \in A$. The relation $\mathop{Ann} A$ is a congruence on the monoid $R$, and $A$ is transformed in a natural fashion into a polygon over the quotient monoid $R/\mathop{Ann} A$. If the polygon $A$ arose from a certain automaton, then this transition is equivalent to identifying identically acting sequences of input signals. In universal algebra one considers the usual constructions of direct and subdirect product, but in addition in polygon theory one may consider a wreath product construction important in the algebraic theory of automata. The free product (or co-product) of polygons coincides with their disjoint union.

A polygon may be regarded as a non-additive analogue of a module over a ring, which serves as a rich source of problems in the theory of polygons. In particular, a relationship has been established between polygons and radicals in semi-groups (cf. Radical in a class of semi-groups), and studies have been made on the relation between the properties of a monoid and those of polygons over them. For example, all left $R$-polygons are projective if and only if $R$ is a one-element group, while the injectivity of all polygons over a commutative monoid $R$ is equivalent to the presence in $R$ of a zero element and the generation of all its ideals by idempotents (see Homological classification of rings).

If $R$ is a monoid with zero 0, one can define an $R$-polygon with zero as a pointed set $A$ which is an $R$-polygon where the distinguished point $u \in A$ satisfies $0a=u$ for all $a\in A$. The theory of polygons with zero has some special features.

Every polygon can be considered as a functor from a one-object category into the category of sets.

In the West, left polygons over a monoid $M$ are usually called $M$-sets; the term "operand" is also in use. The category of all $M$-sets ($M$ fixed) forms a topos; but for this it is essential not to exclude (as above) the empty $M$-set.

Without assuming commutativity as above, the monoids all of whose non-empty left polygons (or, all of whose pointed left polygons) are injective have a few characterizations, which are reviewed in [a3]. As noted above, there are no non-trivial monoids all of whose left polygons are projective, but the perfect monoids, defined (like perfect rings, cf. Perfect ring) by every left polygon having a projective covering, are non-trivial; see [a1], [a2].

The terms monoid action or monoid act, or action of a monoid on a set are also common, as is act for the set acted on; see [b1].

References

[1]	M.A. Arbib (ed.) , Algebraic theory of machines, languages and semigroups , Acad. Press (1968)
[2]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 2 , Amer. Math. Soc. (1967)
[3]	L.A. Skornyakov, "Generalizations of modules" , Modules , 3 , Novosibirsk (1973) pp. 22–27 (In Russian)
[4]	L.A. Skornyakov, A.V. Mikhalev, "Modules" Itogi Nauk. i Tekhn. Alg. Topol. Geom. , 14 (1976) pp. 57–190 (In Russian)
[a1]	J. Fountain, "Perfect semigroups" Proc. Edinburgh Math. Soc. , 20 (1976) pp. 87–93
[a2]	J. Isbell, "Perfect monoids" Semigroup Forum , 2 (1971) pp. 95–118
[a3]	R.W. Yoh, "Congruence relations on left canonic semigroups" Semigroup Forum (1977) pp. 175–183
[a4]	S. Eilenberg, "Automata, languages and machines" , Acad. Press (1974)
[b1]	Mati Kilp, Ulrich Knauer, Alexander V. Mikhalev, Monoids, Acts and Categories: With Applications to Wreath Products and Graphs, Walter de Gruyter (2000) ISBN 3110812908

Density of a sequence

Other definitions

Let $A$ be a set of natural numbers and $\Lambda = (\lambda_n)$ a sequence of weights. The density of $A$ with respect to $\Lambda$ is the limit, if it exists $$ d_\Lambda(A) = \lim_{x \rightarrow \infty} \frac{\sum_{n \le x,\,n \in A} \lambda_n}{\sum_{n \le x} \lambda_n} \ . $$ The upper and lower densities $\bar d_\Lambda$,${\underline d}_\Lambda$ are the corresponding upper and lower limits, if they exist.

The natural or asymptotic density is the density with respect to the constant weight $\lambda_n = 1$ for all $n$.

The logarithmic density is the density with respect to weights $\lambda_n = \frac{1}{n}$.

A sequence has analytic density if and only if it has logarithmic density and if so then they are equal.

References

• Tenenbaum, Gérald, "Introduction to analytic and probabilistic number theory." Cambridge Studies in Advanced Mathematics 46 Cambridge University Press (1995) ISBN 0-521-41261-7 Zbl 0831.11001

Asymptotics of arithmetic functions

asymptotics of number-theoretical functions

Approximate representations of arithmetic functions (functions defined for all natural number values of the argument) by means of comparatively simple expressions with arbitrary small error terms.

An initial question is to ask for a function with the same rate of growth as the given function. Given an arithmetic function $f(x)$ we say that $f$ has rate of growth, or is asymptotic to, a given function $\phi$ if $$ \lim_{x\to\infty}\frac{f(x)}{\phi(x)}=1 \ . $$ A short notation is: $f(x)\sim\phi(x)$ (cf. Asymptotic formula).

Examples:

• The Prime number theorem: $\pi(x)$, the number of primes less than $x$ is asymptotic to $\frac{x}{\log x}$.

More precisely, for an arithmetic function $f(x)$ there exists an asymptotic if one has an asymptotic identity $$ f(x)=\phi(x)+R(x)\,, $$ where $\phi(x)$ is the approximating function or main term, and $R(x)$ is the error term, about which one knows in general only that $$ \lim_{x\to\infty}\frac{R(x)}{\phi(x)}=0 \ . $$ A short notation is: $f(x)=\phi(x)+o(\phi(x))$ or $f(x) = (1+o(1))\phi(x)$.

Examples:

• The Prime number theorem in a more precise form: $\pi(x) = (1+o(1)) \mathop{Li}(x)$ where $\mathop{Li}$ is the logarithmic integral function.

Finding asymptotics of arithmetic functions is one of the most important problems in analytic number theory. This is explained by the fact that most arithmetic functions with interesting arithmetical properties display extreme irregularity in their changes as the argument increases. If one considers instead of an arithmetic function $f(x)$ its average value $(\sum_{n\leq x}f(n))/x$ ($n$ a natural number), then the "irregularity" of $f(x)$ is smoothed out. Hence, a typical problem for an arithmetic function is that of obtaining an asymptotic for its average value function. For example, the average value of the function $\tau(n)$, giving the number of divisors of $n$, is equal to $$ \frac1n\sum_{m\leq n}\tau(m)\sim\ln n. $$ (cf. Divisor_problems#Dirichlet's_divisor_problem). The problem that arises here, of the best possible bound for the error term in the asymptotic identity, is still unsolved (1984) for many functions, in particular for the function $\tau(x)$ (cf. Analytic number theory).

Asymptotics of arithmetic functions play an important a role in additive problems (cf. Additive number theory). For many of them there is no known direct proof of the existence of decompositions of a number into terms of a given form. However, as soon as one has an asymptotic for the number of decompositions of the type one is looking for, one can deduce that the required decomposition exists for all sufficiently large numbers $n$.

The average order and normal order of arithemetic functions refer to some simpler or better-understood function which have comparable asymptotic behaviour. It is conventional to assume that the approximating function $g$ is continuous and monotone.

The average order of an arithmetic function is a function which has the same average.

Let $f$, $g$ be functions on the natural numbers. We say that $f$ has average order $g$ if the asymptotic equality $$ \sum_{n \le x} f(n) \sim \sum_{n \le x} g(n) $$ holds as $x$ tends to infinity.

Examples:

• The average order of $d(n)$, the number of divisors of $n$, is $\log n$;
• The average order of $\sigma(n)$, the sum of divisors of $n$, is $ \frac{\pi^2}{6} n$;
• The average order of $\phi(n)$, the Euler totient function of $n$, is $ \frac{6}{\pi^2} n$;
• The average order of $r(n)$, the number of ways of expressing $n$ as a sum of two squares, is $\pi$;
• The Prime Number Theorem is equivalent to the statement that the von Mangoldt function $\Lambda(n)$ has average order 1.

The normal order of an arithmetic function is a function which "usually" takes the same or closely approximate values.

Let $f$ be a function on the natural numbers. We say that the normal order of $f$ is $g$ if for every $\epsilon > 0$, the inequalities $$ (1-\epsilon) g(n) \le f(n) \le (1+\epsilon) g(n) $$ hold for almost all $n$: that is, the proportion of $n < x$ for which this does not hold tends to 0 as $x$ tends to infinity.

Examples:

• The Hardy–Ramanujan theorem: the normal order of $\omega(n)$, the number of distinct prime factors of $n$, is $\log\log n$;
• The normal order of $\log d(n))$, where $d(n)$ is the number of divisors function of $n$, is $\log 2 \log\log n$.

References

• G.H. Hardy; S. Ramanujan; The normal number of prime factors of a number, Quart. J. Math., 48 (1917), pp. 76–92
• G.H. Hardy; E.M. Wright (2008). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press. ISBN 0-19-921986-5
• Gérald Tenenbaum (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics 46. Cambridge University Press. ISBN 0-521-41261-7