Scholars Constructions of Positive Commutative Semigroups on the Plane, II

. A positive semiroup is a topological semigroup containinq a subseminroup N isomorphic to the multiplicative semigroup of nonnegative real numbers, embedded as a closed subset of E 2 in such a way that is an identity and 0 is a zero. Usina results in Farley [I] it can be shown that positive commutative semigroups on the plane constructed by the techniques given in Farley [2] cannot contain an infinite number of two dimensional groups. In this work an example of such a semigrouD will be constructed which does, however, contain an infinite number of one dimensional groups. Also, some preliminary results are given here concerninQ what types of semilattices of idempotent elements are oossible for positive commutative semigroups on E 2. In particular, we will show that there is a unique positive commutative semigroup on E 2 which is the union of connected groups and which contains five idempotent elements. Also, we will show that such semigroups havinq nine idempotent elements are not unique by constructing an example of such a semigroup with nine idempotent elements whose semilattice of idempotent elements is not "symmetric" and hence which is not isomorphic to the semigroup with nine idempotent elements constructed in Farley [2].

(Received October 2, 1984 and in revised form December lO, 1984) ABSTRACT.A positive semiroup is a topological semigroup containinq a subseminroup N isomorphic to the multiplicative semigroup of nonnegative real numbers, embedded as a closed subset of E 2 in such a way that is an identity and 0 is a zero.Usina results in Farley [I] it can be shown that positive commutative semigroups on the plane constructed by the techniques given in Farley [2] cannot contain an infinite number of two dimensional groups.In this work an example of such a semigrouD will be constructed which does, however, contain an infinite number of one dimensional groups.Also, some preliminary results are given here concerninQ what types of semilattices of idempotent elements are oossible for positive commutative semigroups on E 2. In particular, we will show that there is a unique positive commutative semi- group on E 2 which is the union of connected groups and which contains five idempotent elements.Also, we will show that such semigroups havinq nine idempotent elements are not unique by constructing an example of such a semigroup with nine idempotent elements whose semilattice of idempotent elements is not "symmetric" and hence which is not isomorphic to the semigroup with nine idempotent elements constructed in Farley [2].A topological semigroup is a Hausdorff space together with a continuous, associa- tive multiplication.The author has defined a positive semigroup to be a topological semigroup containing a subsemigroup N isomorphic to the multiplicative semigroup of nonnegative real numbers, embedded as a closed subset of E 2 so that is an identity and 0 is a zero [l].Such semigroups which meet the additional requirement of being the union of groups are called positive Clifford semigroups [I].
In Farley [2] a method was given for constructing positive commutative semigroups on the plane as the union of {0}, an arbitrarily large number of two dimensional groups, and one dimensional groups which are bounding rays of the two dimensional groups.
Since such semigroups are positive commutative Clifford semigroups on E 2, they cannot contain an infinite number of two dimensional groups as shown in Farley [I].However, in this work an example will be constructed of a positive commutative semigroup on E 2 which contains a sector of one dimensional groups, and thus contains an infinite number of one dimensional groups.
While the prob|em of attempting to discover what types of semilattices of idempo- tent elements are possible for positive commutative Clifford semigroups on E 2 appears to be difficult to answer in general, some preliminary results are given in this work.
Using the techniques employed in Farley [I] we will show that there is a unique (up to isomorphism) positive commutative Clifford semigroup on E 2 which is the union of con- nected groups and which contains five idempotent elements.This semigroup, along with its semilattice of idempotent elements will be described herein.Also, we will show that such semigroups having nine idempotent elements are not unique by constructing an example of such a semigroup having nine idempotent elements whose semi lattice of idempotent elenlents is, in sonde sense, not symmetric, and hence is not isomorphic to the semigroup with nine idempotent elements constructed in Farley [2]. 2. PRELIMINARIES.An isomo_rphism_ between two topological semigroups is a function which is both an algebraic isomorphism and a homeomorphism.Unless otherwise indicated, R will denote a semigrouD isomorphic to the multiplicative semigroup of real numbers: N will denote the nonnegative members of R, and P will denote the positive members of R. For a topological space X, we will let {(x,x):x X}.If R is an equivalence relation on X, X/R will denote the set of equivalence classes of X modulo R. A min thread is a closed interval [a,b] together with the usual topology and multiplication defined by x y min{x,y}.

CONSTRUCT ION OF EXAMPLES.
We construct first an example of a positive commutative semigroup on E 2 which contains an infinite number of one dimensional groups.EXAMPLE I. Let us consider two copies of N N denoting them as N N and M M. Also, let us form a sector of one dimensional groups by taking N {min thread} and shrinking the base to a point.Let us call this sector N', and let the idempotent elements on its bounding rays be denoted e 2 and e 3, with e2e3 e 3. Let   [(x,Y)m, (P,e3) c R if and only if x 0 and y p.Let [(s,t)n(P,e2) c R if and only if s O, and t p. Finally, let [(O,ee), (0,0) m] c R and [(O,e), (0,0) n] R.Then, R is clearly an equivalence relation and it is not difficult to check that R is, in fact, a closed congruence.
Thus by [2] S S'/R is a topological semigroup on E2.
It should be noted that an example of this type cannot be constructed using just one copy of N N and sector of one dimensional groups since ele2 would have to be 0 where e (l,O)n.So, this example is more or less minimal.
Upon investigating what types of semilattices of idempotent elements are possible for positive Clifford semigroups on the plane, we first note that the semigroup containing five idempotent elements constructed by the method employed in Example or Example 3 of Farley [2] has the following semilattice of idempotent elements, where denotes the element (1,1)N f denotes (1,1)j, e denotes (I,O)N (l,O)j, e 2 denotes (0,I) N (O,l)j, and 0 denotes (O,O)N (O,O)j" f e e .
For positive Clifford seT,groups on E 2 which are the unon of connected groups, we wl show that the semgroup s unique (up to isomorphism) among those having exactly five dempotent elements.
THEOREM 1.The positive commutative Clifford seT, group S on E 2 which contains exactly fve idenpotent elements and is the union of connected groups is unique.
PROOF" Let H(1) denote the group having the identity eement l, and let denote the closure of H(1).Then gL(EH(1)] s isomorphic to N N as shown n Farley [1], since CLEH(I)] is two dimensional by Horne [3].Let e and e 2 denote the idempo- tent elements on Pe and Pe 2, the bounding rays of H(1), let f be the remaining idempo- tent element which is in E2\CL[H(1)], and let H(f) be the group with identity element f.Ther, the only possibilities for the semilattice of idempotent elements are the one given above Fi,Lre l) and the other iven in Figure 2.
We now show that this semilattice is, in fact, impossible.Since S has exactly five idempotent elements and is the union of connected groups, H(f) E2CL[H(1)].Now, suppose e If O. Let {X n} be a sequence in H(f) such that {X n} e I. Then {elXn}-e e But, elX el(fX), since f is the identity element in H(f).So, {el(fXn)} e However, each term of the sequence {el(fXn)} is O, under the assump- tion that elf O.This implies {el(fXn)} 0 which is a contradiction.
A positive Clifford semigroup having nine idempotent elements as constructed by the method employed in Example 3 of Farley [2] has the following semi lattice of idempo- tent elements, where denotes (l,l)n, j denotes (l,l)j, m denotes (l,l)m, k denotes (l,l) k, e denotes (l,O) n (l,O)j, g denotes (O,l)j (l,O)m, h denotes (O,l)m (l,O) k, f denotes (O,l)k (O,l)n and 0 denotes (O,O)m (O,O)j (0,0) n (O,O)k'(Fig. 3):  We no construct an exanple of a positive Clifford semgroup on E g ose semi- lattice of nine idempotent elements s not {somorpc to te one bove, and some sense, not symmetr{c.(a,b)m (M.M), (0 0), and (a,b)k (K K). lis multiplication {s easily verified to be continuous and associative.So, it te usual topology, S' s a topological semigroup, ket us define a relat{on R on S' s follows, ket R, and let R be syetec by definition.R is easily checked to be a closed congruence, and consequently S'/R s a topological semgroup on E g. ket us denote by e the element (1,O)n (] ,0) 3, by 9 the element (0,1) 3 (0,1) m, by te element (1,O)m (1,0)k and by f the element (0,1)k (0,1)n.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: (a,b) n denote an element of N N, let (a,b)m denote an element of M M, and let (p, e denote an element of N', where e is an idempotent element.Let us define a multiplication on S' [(N N) U (M M) U N'] in the following manner.Let multiplication be coordi- natewise in N N, M M, and N'.Let us define (a,b)n (p,e) (bp,e) (p,e).(a,b)n, (P'e) (p',e_.)=(pp',ee)= (p',e)-(p,e), (a,b)n (c d)m (ac,bd) m (c,d) m (a,b) n, and (p,e) (x,Y)m (O,py)m (x'Y)m (p,e).Thismultiplication is easily checked to be associative, and its continuity follows from the continuity of multiplication in N N, M M, and N'.So, with the usual topology, S' is a topological semigroup.Let us define a relation R on S' in the following way.Let R, and let R be symmetric by definition.Let [(a,b)m,(C,d)n] R if and only if a c, and b 0 d.Let Figure 3.

EXAMPLE 2 .
ket us consider four copes of N N. ket us denote these copies as N N (wicn {11 be g), 0 O, M M, and K K. ket us define 8 mult{plication on S' [( N) O (0 O) O (M M) U (K K)] n te followin9 manner, ket the multi- plication be coordJnatese in each copy of N N, and let te multiplication be commutative on a11 of S'. ket (s,b)j (c,d)m (c,bd)m,(a,b) (c,d) (ac,bCd)m (a,b) k (c,d) m (ac,abd),,, (a,b)n (c,d) (ac,abd)j, (a,b)n (c,d)m (ac,aDd)m and (a,b)n (c,d)k (ac,bd)k ere (a,b)n ( N), ket [(a,b)j,(c,d)n R if nd only if a c and b 0 d.ket [(a,b)j,(c,d)m and only if b d and a 0 c. ket [(a,b)m,(C,d)n R if and only if 8 c and b 0 d.Finally, let [(a,b)k,(c,d)n R {f and only if b d and a 0 c.

First
Round of Reviews May 1, 2009 KEY WORDS AND PHRASES.Positive semigroup, Clifford semi.group,semiattioe of idempotent e ements.