A note on the problem of the parallelogram of forces (the axiomatization of vector addition)

In this chapter we deal with semigroups which satisfy the identity axy = ayx. These semigroups are called right commutative semigroups. It is clear that a right commutative semigroup is medial and so we can use the results of the previous chapter for right commutative semigroups. For example, every right commutative semigroup is a semilattice of right commutative archimedean semigroups and is a band of right commutative t-archimedean semigroups. A semigroup is right commutative and simple if and only if it is a left abelian group. Moreover, a semigroup is right commutative and archimedean containing at least one idempotent element if and only if it is a retract extension of a left abelian group by a right commutative nil semigroup. We characterize the right commutative left cancellative and the right commutative right cancellative semigroups, respectively. Clearly, a semigroup is right commutative and left cancellative if and only if it is a commutative cancellative semigroup. A semigroup is right commutative and right cancellative if and only if it is embeddable into a left abelian group if and only if it is a left zero semigroup of commutative cancellative semigroups. It is shown that a right commutative semigroup is embeddable into a semigroup which is a union of groups if and only if it is right separative.

Every commutative semigroup is a semilattice of archimedean subsemigroups. Given a semilattice Y and a collection of commutative semigroups S indexed by Y, the converse problem of constructing every commutative semigroup which is a semilattice (Y) of the semigroups S remains unsolved except in certain cases. If each S is an abelian group the result follows as a special case of Clifford's Theorem (see [33, Thm. 4.11). The authors [13 have given a solution for any semilattice where each S is assumed to be cyclic, and more recently, Tamura has solved the problem for exclusive semilattices of exclusive semigroups [63 For a characterization of the general semilattice decomposition when each Se is weakly reductive, see Theorem 7.8.13 of [41. An element x of a commutative semigroup S is said to be prime if x ~ S 2 . S is said to have unique factorization if each nonzero element of S can be written uniquely as a product of primes in the usual sense. A commutative semigroup has unique factorization if and only if it is free or a Rees quotient of a free commutative semigroup (E2~, Thm. i). Commutative nil unique factorization semigroups were used in the characterization of arbitrary commutative nil semigroups in [11 (see [53 for an alternate characterization). Since commutative unique factorization semigroups arise naturally in the general theory of commutative semigroups, it seems of interest

In this chapter we deal with semigroups which are both ∛-commutative and conditionally commutative. These semigroups are called ∛C-commutative semigroups. The ∛-commutative semigroups and the conditionally commutative semigroups are examined in Chapter 5 and Chapter 6, respectively. From the results of those chapters it follows that every C-commutative semigroup is a semilattice of conditionally commutative a∛Chimedean semigroups. In this chapter, we show that the simple ∛C-commutative semigroups are exactly the right abelian groups. By the help of this result we show that every ∛C-commutative a∛Chimedean semigroup containing at least one idempotent element is an ideal extension of a right abelian group by a commutative nil semigroup. As a consequence, we prove that every ∛C-commutative regular semigroup is a spined product of a right normal band and a semilattice of abelian groups. We determine the subdirectly irreducible ∛C-commutative semigroups with a globally idempotent core. We show that they are those semigroups which are isomorphic to either G or G0 or F or R or R0, where G is a non-trivial subgroup of a quasicyclic p-group (p is a prime), F is a two-element semilattice and R is a two-element right zero semigroup. At the end of the chapter we deal with the ∛C-commutative G0-semigroups. It is shown that a semigroup S is an Ccommutative d-semigroup if and only if it is isomorphic to either G or G0 or R or R0 or N or N 1 , where G is a non-trivial subgroup of a quasicyclic p-group (p is a prime), R is a two-element right zero semigroup and N is a commutative nil semigroup whose ideals form a chain with respect to inclusion.

In an earlier paper we defined and proved some properties of the tensor product of semigroups. It turns out that the category of commutative semigroups has also a tensor product, which is in many respects more interesting. It keeps all the main properties of the tensor product of arbitrary semigroups (preservation of one-to-one consistent homomorphisms, right exactness, adjoint associativity) and is furthermore colimit-preserving and an associative operation; when applied to abelian groups, it gives the ordinary tensor product of groups. These properties occupy the second section of this paper, the first being devoted to some examples. In the third section, we study flat commutative semigroups. We prove that a flat commutative semigroup with a minimal generating subset must be free, but that flatness is not hereditary and that there exist flat commutative semigroups which are not free. The nature of flat commutative semigroups remains open, as well as the noncommutative case. Heavy use is made of the properties of the tensor product of arbitrary semigroups, which we established in [5]. For the fundamentals of semigroup theory, the reader is referred to [1] and [2]. Throughout, the largest commutative homomorphic image of a semigroup A will be denoted by C(A); the largest idempotent (resp. normal) homomorphic image of A will be denoted by E(A) (resp. N(A)) (normal means: (xy)n = xyn for all x, y E A and all n [5], [8]). These are covariant functors, in fact coreftexions, of the category of all semigroups onto the full subcategory under consideration; as coreflexions, they preserve colimits and, for instance, the semigroups Hom (A, B) and Hom (C(A), B) are naturally isomorphic whenever B is commutative (if B is commutative, Hom (A, B) is the (commutative) semigroup of all homomorphisms of A into B with pointwise multiplication; for coreflexions, see [6]).

Subdirectly irreducible universal algebras, i.e., algebras which are not decomposable into non-trivial subdirect product, play an important role in mathematics due to well-known Birkhoff Theorem which states that any algebra is a subdirect product of subdirectly irreducible algebras (in another terminology: every algebra is approximated by the subdirectly irreducible algebras). In view of this, it seems reasonable to study algebras with certain conditions on subdirectly irreducible algebras. One of the natural restrictions is the finiteness of all subdirectly irreducible algebras. More stronger restriction is boundness of orders of all subdirectly irreducible algebras. An act over a semigroup (it is also an automaton, and a unary algebra) is a set with an action of the given semigroup on it. The acts over a fixed semigroup form a variety whose signature coincides with self semigroup. On the other hand, it is a category whose morphisms are homomorphisms from one act into another. It is not difficult to see that the semigroups over which all subdirectly ireducible acts are finite, are exactly the semigroups over which all subdirectly irreducible acts are finitely approximated (in another terminology: residually finite). A more narrow class form semigroups over which all acts are approximated by acts of n or less elements where n is a fixed natural number. In 2000, I.B.Kozhukhov proved that all non-trivial acts over a semigroup S are approximated by two-element ones if and only if S is a semilattice (a commutative idempotent semigroup). In 2014, I.B.Kozhukhov and A.R.Haliullina proved that any semigroup with bounded orders of dubdirectly irreducible acts is uniformly locally finite, i.e., for every k, the orders of k-generated subsemigroups are bounded. In the work of I.B.Kozhukhov and A.V.Tsarev 2019, the authors described completely the abelian groups over which all acts are finitely approximated, and also abelian groups over which all acts are approximated by acts of bounded orders. In this work, we characterize the commutative semigroups over which all acts are approximated by acts consisted of n or less elements.

The main concern is the existence of a maximal compact normal subgroup K in a locally compact group G, and whether or not G/K is a Lie group. G has a maximal compact subgroup if and only if G/Go has. Maximal compact subgroups of totally disconnected groups are open. If the bounded part of G is compactly generated, then G has a maximal compact normal subgroup K and if B(G) is open, then G/K is Lie. Generalized FCgroups, compactly generated type I IN-groups, and Moore groups share the same properties. Introduction. We shall be concerned with the conditions under which a locally compact group G possesses a maximal compact normal subgroup K and whether or not G/K is a Lie group. It is known that locally compact connected groups and compactly generated Abelian groups have maximal compact normal subgroups with Lie factor groups. Moreover, almost connected groups (i.e., when G/Go is compact, where Go is the identity component of G) and compactly generated FCgroups share these results. For general locally compact Abelian groups, the corresponding assertion is not true even if the group is a-compact. Consider any Q,a of a-adic numbes; Q7a is a locally compact a-compact Abelian group which has no maximal compact (normal) subgroup. In this paper we show that a locally compact group G has a maximal compact subgroup if and only if G/Go has (Theorem 1). This generalizes substantially the corresponding result for almost connected groups mentioned before. It also reduces the question of the existence of maximal compact subgroups to that of totally disconnected groups-in which we show every maximal compact subgroup is open (Corollary 1 of Theorem 2). In [1] it is proved that a generalized FCgroup G has a maximal compact normal subgroup K. Here we show that G/K is a Lie group (Theorem 5). We also prove that, if the bounded part of G, B(G), is compactly generated, then G has a maximal compact normal subgroup (Theorem 6). Particularly, we see that compactly generated Moore groups, type I IN-groups, and compact extensions of compactly generated nilpotent groups have the desired properties (Theorems 7 and 8). Finally some sufficient conditions for G/K to be a Lie group are summarized in Theorem 10. If a topological group G has a maximal compact subgroup N, then clearly gNgis a maximal compact subgroup of G for any g C G. A maximal compact subgroup contains every compact normal subgroup. Hence K = ngeG gNg-1 is the maximal compact normal subgroup of G. Received by the editors July 12, 1985 and, in revised form, December 24, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 22D05. (?1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

This chapter gives a brief introduction to the mathematics involved in the determination of the subgroups of space groups. The algebraic concepts of vector spaces, the affine space and the affine group are defined and discussed. A section on groups with special emphasis on actions of groups on sets, the Sylow theorems and the isomorphism theorems follows. After the definition of space groups, their maximal subgroups are considered and the theorem of Hermann is derived. It is shown that a maximal subgroup of a space group has a finite index and is a space group again. From the proof that three-dimensional space groups are soluble groups, it follows that the indices of their maximal subgroups are prime powers. Special considerations are devoted to the subgroups of index 2 and 3. Furthermore, a maximal subgroup is an isomorphic subgroup if its index is larger than 4. In addition, more special quantitative results on the numbers and indices of maximal subgroups of space groups are derived. The abstract definitions and theorems are illustrated by several examples and applications. Keywords: Abelian groups; CARAT; Euclidean affine spaces; Euclidean groups; Euclidean metrics; Euclidean vector spaces; Hermann’s theorem; Lagrange’s theorem; Sylow’s theorems; affine groups; affine mappings; affine spaces; automorphism; characteristic subgroups; congruence; factor groups; faithful actions; isomorphism; generators; groups; homomorphism; klassengleiche subgroups; cosets; isomorphic groups; mappings; normalizers; space groups; isomorphism theorems; translation subgroups; maximal subgroups; translationengleiche subgroups; vector spaces; orbits; alternating groups; symmorphic space groups

This chapter gives a brief introduction to the mathematics involved in the determination of the subgroups of space groups. The algebraic concepts of vector spaces, the affine space and the affine group are defined and discussed. A section on groups with special emphasis on actions of groups on sets, the Sylow theorems and the isomorphism theorems follows. After the definition of space groups, their maximal subgroups are considered and the theorem of Hermann is derived. It is shown that a maximal subgroup of a space group has a finite index and is a space group again. From the proof that three-dimensional space groups are soluble groups, it follows that the indices of their maximal subgroups are prime powers. Special considerations are devoted to the subgroups of index 2 and 3. Furthermore, a maximal subgroup is an isomorphic subgroup if its index is larger than 4. In addition, more special quantitative results on the numbers and indices of maximal subgroups of space groups are derived. The abstract definitions and theorems are illustrated by several examples and applications. Keywords: Abelian groups; CARAT; Euclidean affine spaces; Euclidean groups; Euclidean metrics; Euclidean vector spaces; Hermann’s theorem; Lagrange’s theorem; Sylow’s theorems; affine groups; affine mappings; affine spaces; automorphism; characteristic subgroups; congruence; factor groups; faithful actions; isomorphism; generators; groups; homomorphism; klassengleiche subgroups; cosets; isomorphic groups; mappings; normalizers; space groups; isomorphism theorems; translation subgroups; maximal subgroups; translationengleiche subgroups; vector spaces; orbits; alternating groups; symmorphic space groups

Abstract: In this work, the set of all functions that are Laplace transformable with regard to their structures both algebraic and topological, is taken into account. Certain topological properties of the set of Laplace transformable functions with the help of a metric are described. Also, we determine the proofs of the statements that the set of all Laplace transformable functions is a commutative semi-group with respect to the convolution operation as well as an Abelian group with respect to the operation of addition. Metric for two functions belonging to the set of all Laplace transformable functions is defined and the proof that the Laplace transformable functions' space is complete with our metric is given. The separability theorem and that the Laplace transformable functions' space is disconnected are also discussed. Keywords: Abelian group, Commutative semi-group, Disconnected space, Laplace transform, Separability theorem.

Abasefor a commutative semigroup (S, +) is an indexed set 〈xt〉t∈AinSsuch that each elementx∈Sis uniquely representable as Σt∈FxtwhereFis a finite subset ofAand, ifShas an identity 0, then 0 = Σn∈Øxt. We investigate those commutative semigroups or groups which have a base. We obtain the surprising result thathas a base. More generally, we show that an abelian group has a base if and only if it has no elements of odd finite order.

Although the general solution of the cocycle functional equation on abelian groups is well-known, the theory concerning solutions of that equation on commutative semigroups is far from complete. Here we introduce a framework which allows us to make a significant advance in the development of this theory.

Ideal extensions of semigroups were introduced by Clifford in [1] and since then they have been widely studied (see for instance [2]). Our aim here is to characterize commutative ideal extensions of Abelian groups. We show that they are those commutative semigroups with an idempotent Archimedean element, or equivalently, those commutative semigroups E such that E/R is an Abelian group, where R is the least congruence that makes E/R cancellative. These characterizations give rise to an algorithm for deciding from a presentation of a finitely generated commutative monoid whether it is an ideal extension of an Abelian group. Finally we present a procedure that enables us to compute the set of idempotents of a finitely generated commutative monoid. All semigroups appearing in this paper are commutative and for this reason in the sequel we sometimes omit the adjective commutative whenever we refer to a commutative semigroup (the same holds for monoids). The authors would like to thank the referee for his/her comments and suggestions.

In this chapter we deal with semigroups satisfying the identity axb = bxa. These semigroups are called externally commutative semigroups. It is clear that an externally commutative semigroup is medial. Thus the externally commutative semigroups are semilattice of externally commutative archimedean semigroups. A semigroup is externally commutative and 0-simple if and only if it is a commutative group with a zero adjoined. A semigroup is externally commutative and archimedean containing at least one idempotent element if and only if it is an ideal extension of a commutative group by an externally commutative nil semigroup. Moreover, every externally commutative archimedean semigroup without idempotent has a non-trivial group homomorphic image. We show that an externally commutative semigroup is regular if and only if it is a semilattice of commutative groups. We construct the least separative, left separative, right separative and weakly separative congruence on an externally commutative semigroup, respectively. We determine the subdirectly irreducible externally commutative semigroups. We prove that a semigroup is subdirectly irreducible and externally commutative with a globally idempotent core if and only if it is isomorphic to either G or G 0 or F, where G is a non-trivial subgroup of a quasicyclic p-group (p is a prime) and F is a two-element sernilattice. An externally commutative semigroup with a zero and a non-trivial annihilator is subbdirectly irreduucible if and only if it has a non-zero disjunctive element.

The construction of the integers introduced by Dedekind is an algebraic one. Subtraction can not be done without restrictionin natural numbers N. If we consider the definition of multiplication of integral domain Z, N with respect tosubtraction is needed. It is necessary to give the definition of subtraction in N. Instead of starting from natural numbers,one could begin with any commutative semi-group and construct from it as the construction of the integers to obtain acommutative group. If the cancellation law does not hold in the commutative semi-group, some modifications are required.The mapping from the commutative semi-group to the commutative group is not injective and compatible withaddition. In the relation between real numbers and decimals, N also plays an important role.

Some thirty-five years ago, Suprunenko proved that every symmetric group S~ possesses a unique conjugacy class of maximal (with respect to inclusion) nilpotent transitive subgroups, and also the intransitive maximal nilpotent subgroups of Sn can be classified (see [3]). The purpose of this paper is the classification of all maximal supersoluble subgroups in symmetric groups. In contrast to the nilpotent case, there is usually more than one conjugacy class of maximal supersoluble transitive subgroups in S~; even more, there is no bound (independent of n) on the number of those conjugacy classes. The proof of the main result in this paper splits into two parts. In Section 4 we present a certain general construction of maximal supersoluble transitive subgroups of a given symmetric group Sn and in Section 5 we show that there are no others. The construction depends on certain ,~admissible, factorizations of n and on the choice of a certain regular abelian group. Different admissible factorization of n yield different conjugacy classes of maximal supersoluble transitive subgroups of S~; whether there are any non-trivial admissible factorizations of n (the trivial ones are n = n. 1 if n is odd and n = (n/2). 2 if n is even) depends on the primes involved in n. Exactly when there are odd prime divisors p and q of n such that q divides p 1 or when 4 divides n and n is not a power of 2, there exist a non-trivial admissible factori-