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Research article Special Issues

Non-singular solutions of p-Laplace problems, allowing multiple changes of sign in the nonlinearity

  • Received: 20 September 2021 Revised: 30 November 2021 Accepted: 30 November 2021 Published: 18 March 2022
  • For the p-Laplace Dirichlet problem (where φ(t)=t|t|p2, p>1)

    φ(u(x))+f(u(x))=0for1<x<1,u(1)=u(1)=0

    assume that f(u)>(p1)f(u)u>0 for u>γ>0, while γuf(t)dt<0 for all u(0,γ). Then any positive solution, with max(1,1)u(x)=u(0)>γ, is non-singular, no matter how many times f(u) changes sign on (0,γ). The uniqueness of solution follows.

    Citation: Philip Korman. Non-singular solutions of p-Laplace problems, allowing multiple changes of sign in the nonlinearity[J]. Electronic Research Archive, 2022, 30(4): 1414-1418. doi: 10.3934/era.2022073

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  • For the p-Laplace Dirichlet problem (where φ(t)=t|t|p2, p>1)

    φ(u(x))+f(u(x))=0for1<x<1,u(1)=u(1)=0

    assume that f(u)>(p1)f(u)u>0 for u>γ>0, while γuf(t)dt<0 for all u(0,γ). Then any positive solution, with max(1,1)u(x)=u(0)>γ, is non-singular, no matter how many times f(u) changes sign on (0,γ). The uniqueness of solution follows.



    Primitive semigroups have been an important topic of semigroup researches since the 1950s. In fact, as early as 1954, Preston who is one of the founders of the algebraic theory of semigroups studied primitive inverse semigroups in [16], and then he gave the characterization of primitive regular semigroups and applied it to the study of matrix representations of inverse semigroups in [17]. In particular, he pointed out that a regular semigroup (resp. an inverse semigroup) with zero is primitive if and only if it is the 0-direct union of a family of completely 0-simple semigroups (resp. Brandt semigroups) (See also the monographs [5,12,18]). In addition, from the view of category Lawson [12] showed that an inverse semigroup with zero is primitive if and only if it is isomorphic to a groupoid with zero adjoined. On the other hand, Theorem Ⅲ.3.5 and Corollary Ⅲ.3.6 in [5] together give that a regular semigroup (resp. an inverse semigroup) without zero is primitive if and only if it is a completely simple semigroup (resp. a group). Furthermore, primitive orthodox semigroups were also concerned in Venkatesan [21].

    As generalizations of regular semigroups, abundant semigroups were introduced and investigated in 1982 by Fountain in [2] where the class of primitive abundant semigroups and its several subclasses, such as primitive abundant semigroups with regularity condition, primitive quasi-adequate semigroups and primitive adequate semigroups etc., were also characterized. We observe that the roles of quasi-adequate semigroups and adequate semigroups in the range of abundant semigroups are similar to those of orthodox semigroups and inverse semigroups in the range of regular semigroups, respectively.

    In 1991, Lawson [13] went a further step to generalize abundant semigroups to U-semiabundant semigroups where U is a nonempty subset of the set of idempotents and correspondingly generalize quasi-adequate semigroups and adequate semigroups to weakly U-orthodox semigroups and Ehresmann semigroups, respectively. The class of Ehresmann semigroups and its special subclasses (for example, the class of restriction semigroups) now form a hot research topic, and a lot of achievements in this line have been obtained by many semigroup experts, for example, see [3,4,6,8,10,11,13,20] and the references therein. In particular, Gould [3,4] gave the equivalent definition of Ehresmann semigroups from the view of variety, and Jones explicitly introduced the notion of primitive Ehresmann semigroups in [8] and by using small categories obtained a construction of primitive Ehresmann semigroups with zero in [10] which is analogous to that of primitive inverse semigroups with zero given in Lawson [12] by using groupoids. We also observe that Lawson [14] investigated a class of primitive U-semiabundant semigroups named Rees semigroups and Wang [22] characterized primitive weakly U-orthodox semigroups, which generalize the corresponding results of primitive abundant semigroups provided in [2].

    On the other hand, Jones [7] generalized Ehresmann semigroups to P-Ehresmann semigroups from a varietal perspective and provided a common framework for Ehresmann semigroups and regular *-semigroups. Regular *-semigroups first appeared in Nordahl and Scheiblich [15] and a generalization of this class of semigroups was investigated in the author [23]. For more details for regular -semigroups, the reader may consult the texts[1,7,15,19,23] and their references. At present, some valuable results have been obtained on P-Ehresmann semigroups. For instance, the constructions of P-Ehresmann semigroups have been considered by "fundamental approach" in [7] and [25,26] by "categorical approach", respectively. Variety properties, semigroup algebras and completions of P-Ehresmann semigroups have been explored in Jones [8], Wang [24] and Yan and Wang [27], respectively.

    From the above discussions, the following problem is natural: How to introduce and characterize primitive P-Ehresmann semigroups? The aim of this paper is to solve the above problem. We have introduced the notion of projection-primitive P-Ehresmann semigroups and established the structures of projection-primitive P-Ehresmann semigroups. In particular, we show that projection-primitive P-Ehresmann semigroups are always P-restriction. Our work may be regarded as extending primitive Ehresmann semigroups introduced and investigated by Jones in [8] and [10], respectively.

    In this section, after recalling some necessary notions and results on P-Ehresmann semigroups, we shall introduce projection-primitive P-Ehresmann semigroups and explore their basic properties.

    For a semigroup S, we always denote the set of idempotents in S by E(S). From Lemma 2.2 and its dual in Gould [3], a bi-unary semigroup (S,,+,) is called an Ehresmann semigroup if the following identities hold:

    x+x=x,(xy)+=(xy+)+,(x+y+)+=x+y+,x+y+=y+x+,(x+)=x+
    xx=x,(xy)=(xy),(xy)=xy,xy=yx,(x)+=x.

    To extend Ehresmann semigroups, P-Ehresmann semigroups were introduced in Jones [7] from the view of variety. A bi-unary semigroup (S,,+,) is called a P-Ehresmann semigroup if the following identities hold:

    A P-Ehresmann semigroup (S,,+,) is called P-restriction if

    (xy)+x=xy+x and x(yx)=x+yx for all x,yS.

    In a P-Ehresmann semigroup (S,,+,), the set of projections is PS={a+aS} which is equal to {aaS} by (ⅴ) and (ⅴ). The following lemmas collect some basic properties of P-Ehresmann semigroups first given in Jones [7].

    Lemma 2.1 ([7]). A bi-unary semigroup (S,,+,) is Ehresmann (resp. restriction) if and only if (S,,+,) is P-Ehresmann (resp. P-restriction) and PS is a subsemilattice of S.

    Lemma 2.2 ([7]). Let (S,,+,) be a P-Ehresmann semigroup and x,yS,e,fPS.

    (1) (x+y)+=x+y+x+,x++=x+,x+(xy)+x+=(xy)+.

    (2) (xy)=yxy,x=x,y(xy)y=(xy).

    (3) (ef)2=ef,e+=e=e,(ef)+=efe=(fe)PS.

    (4) efPS if and only if ef=fe.

    Let (S,,+,) be a P-Ehresmann semigroup. Define a relation on PS by the rule

    efifandonlyife=ef=feforalle,fPS.

    Then it is easy to see that is a partial order on PS. By Lemma 2.2, the following corollary is obvious.

    Corollary 2.3. Let (S,,+,) be a P-Ehresmann semigroup and x,yS,e,fPS. Then (xy)+x+,(xy)y and efee.

    Similar to the case of restriction semigroups appeared in Jones [8], we call a P-Ehresmann semigroup (S,,+,) without zero or with zero 0 satisfying 0PS projection-primitive if

    (e,fPS)efe=f,

    while call a P-Ehresmann semigroup (S,,+,) with zero 0 satisfying 0PS projection-primitive if

    (e,fPS)efe=0ore=f.

    Observe that a primitive Ehresmann semigroup without zero is just a momoid in which the identity is the only projection by Lemma 2.1. We first characterize projection-primitive P-Ehresmann semigroups without zero.

    Proposition 2.4. Let (S,,+,) be a P-Ehresmann semigroup without zero. Then the following statements are equivalent:

    (1) S is projection-primitive.

    (2) (xy)+=x+ for all x,yS.

    (3) (xy)=y for all x,yS.

    Proof. We only show that (1) is equivalent to (2), and one can prove that (1) is equivalent to (3) by similar arguments. If S is projection-primitive and x,yS, then (xy)+x+ by Corollary 2.3. This implies that (xy)+=x+. Conversely, let e,fPS and ef. Then ef=fe=e. By the given condition and Lemma 2.2, we have f=f+=(fe)+=e+=e. This gives the projection-primitivity of S.

    From Jones [9], a P-Ehresmann semigroup (S,,+,) is called reduced if PS contains exactly one element. By the identities (ⅰ) and (ⅰ), in this case S is a monoid with the only projection as its identity. Obviously, reduced P-Ehresmann semigroups are always projection-primitive. In fact, we have the following result.

    Proposition 2.5. Let (S,,+,) be a P-Ehresmann semigroup with zero 0 and 0PS. Then the following statements are equivalent:

    (1) S is projection-primitive.

    (2) x+=0+ for all xS.

    (3) x=0 for all xS.

    In this case, S is a reduced P-Ehresmann semigroup with the identity 0+ and so is an Ehresmann semigroup.

    Proof. We only show that (1) is equivalent to (2), and one can prove that (1) is equivalent to (3) by similar arguments. Let S be projection-primitive and xS. Then 0+=(x0)+x+ by Corollary 2.3, and so 0+=x+. Conversely, the given condition (2) implies that PS={x+xS}={0+}. This gives that S is reduced and has identity 0+, and so is projection-primitive certainly.

    Remark 2.6. By Proposition 2.5, a projection-primitive P-Ehresmann (or Ehresmann) semigroup (S,,+,) with zero 0 and 0PS is reduced and is exactly a monoid with zero containing at least two elements.

    Now we consider projection-primitive P-Ehresmann semigroups with zero 0 as a projection.

    Proposition 2.7. Let (S,,+,) be a projection-primitive P-Ehresmann semigroup with zero 0 and 0PS.

    (1) For all xS, x+=0x=0x=0.

    (2) For all x,yS{0}, xy0xy+x=xy+xy+=y+.

    Proof. (1) Let xS. We only prove that x+=0 if and only if x=0. The other equivalence can be showed by symmetry. In fact, if x+=0, then x=x+x=0x=0 by (ⅰ). To show the converse, we first observe that 0+=(a0)+a+ for all aS by Corollary 2.3. This implies that 0+ is the minimum element in PS. Since 0PS, it follows that 0+=0+0=0.

    (2) Let x,yS{0}. We only prove that xy0 if and only if xy+x=x. The other equivalence can be showed by symmetry. If xy0, then xxy+y=xy0 by the identities (ⅰ) and (ⅰ). This implies that x0 and xy+0. By item (1) we have x+0 and (xy+)+0. But Corollary 2.3 gives that (xy+)+x+, and so (xy+)+=x+ by the projection-primitivity of S. Moreover, Lemma 2.2 provides that xy+x=(xy+)+=x+=x. Conversely, if xy+x=x, then by using (ⅱ), (ⅰ), (ⅱ), Lemma 2.2 (3), (ⅰ) and item (1) of the present lemma in order, we have

    (xy)+=(xy+)+=(xxy+)+=(x(xy+)+)+=(xxy+x)+=(xx)+=x+0, (2.1)

    which implies that xy0 by item (1) of the present lemma again. To end this section, we observe the following interesting result.

    Theorem 2.8. A projection-primitive P-Ehresmann semigroup is always P-restriction.

    Proof. Let (S,,+,) be a projection-primitive P-Ehresmann semigroup and x,yS. Firstly, if S contains no zero, then (xy)+x=x+x=x by Proposition 2.4. By Corollary 2.3, we have xy+xx, and so xy+x=x by the projection-primitivity of S. This implies that xy+x=xxy+x=xx=x by (ⅰ). Thus (xy)+x=xy+x. Dually, we have x(yx)=x+yx. Secondly, if S contains a zero 0 and 0PS, then it is obvious that (xy)+x=xy+x and x(yx)=x+yx by Proposition 2.5. Finally, assume that S contains a zero 0 and 0PS. If xy=0, then by Proposition 2.7 (1) and (ⅱ) we have 0=(xy)+=(xy+)+ and xy+=0. This implies that (xy)+x=0=xy+x. If xy0, then (xy)+0 and x=xy+x by Proposition 2.7. Since (xy)+x+ by Corollary 2.3, the projection-primitivity of S gives that (xy)+=x+. This implies that

    xy+x=xxy+x=xx=x=x+x=(xy)+x

    by (ⅰ) and (ⅰ). Therefore, (xy)+x=xy+x. Dually, we have x(yx)=x+yx. Thus, S is P-restriction in all cases.

    In the remainder of this paper, we shall establish the structures of projection-primitive P-Ehresmann semigroups. The present section is devoted to projection-primitive P-Ehresmann semigroups without zero or with zero which is not a projection. The following theorem characterize these semigroups completely.

    Theorem 3.1. Let I and Λ be two sets and ϕ:IΛ,iiϕ be a bijection. Assume that M is a monoid, |I×M×Λ|1 and P=(pλi)Λ×I is a Λ×I-matrix over M satisfying piϕ,i=e=piϕ,jpjϕ,i for all i,jI, where e is the identity of M. Define a binary and two unary operations on the set

    S=M(I,Λ,M,P)={(i,x,λ)iI,xM,λΛ}

    as follows:

    (i,x,λ)(j,y,μ)=(i,xpλjy,μ),(i,x,λ)+=(i,e,iϕ),(i,x,λ)=(λϕ1,e,λ).

    Then (S,,+,) is a projection-primitive P-Ehresmann semigroup without zero or with zero which is not a projection. Conversely, every such semigroup can be obtained in this way.

    Proof. Direct part. By hypothesis, S0 can be regarded as a Rees matrix semigroup over the monoid M0. Denote P(S0)={(i,p1λi,λ)iI,λΛ}{0} and U={(i,e,iϕ)iI}. Then by Proposition 1.5 in Lawson [14], we can easily show that for all (i,x,λ),(j,y,μ)S,

    (i,x,λ)˜RP(S0)(j,y,μ)(i,x,λ)˜RU(j,y,μ)i=j,
    (i,x,λ)˜LP(S0)(j,y,μ)(i,x,λ)˜LU(j,y,μ)λ=μ.

    Moreover, ˜RU (resp. ˜LU) is a left congruence (resp. a right congruence) on S by Lemma 1.9 in Lawson [14]. By the given condition on the matrix P, it is easy to see that U is both a right projection-set and a left projection-set of S in the sense of Jones [7] (see page 629). According to Theorem 6.1 of Jones [7] and its dual, (S,,+,) is a P-Ehresmann semigroup and PS=U. Moreover, in view of Proposition 1.7 of Lawson [14], no two different elements in PS can be compatible. Thus S is projection-primitive. If (i,z,λ) is the zero element of S, then for all (j,y,μ)S, we have (i,z,λ)(j,y,μ)=(i,z,λ)=(j,y,μ)(i,z,λ). This implies that i=j and λ=μ. In this case, |I|=|Λ|=1 and so PS contains only one element. By hypothesis, S has at least two elements and so the zero is not a projection.

    Converse part. Let (M,,+,) be a projection-primitive P-Ehresmann semigroup with zero which is not a projection. Then by Proposition 2.5 and Remark 2.6, M has at least two elements and is a monoid with the only projection e as its identity. In this case, M has the form in the theorem certainly.

    Now let (T,,+,) be a projection-primitive P-Ehresmann semigroup without zero. Then it is easy to see that T0 is a Rees semigroup with respect to U=P0T in the sense of Lawson [14], and ˜LU={(a,b)T0×T0a=b}{(0,0)} and ˜RU={(a,b)T0×T0a+=b+}{(0,0)} (see page 28 in [14]). Fix an element ePT and denote I={xexPT},Λ={exxPT}. Define ϕ:IΛ,xeex for all xPT. Then ϕ is a bijection. In fact, if x,yPT and xe=ye, then x=x+=(xe)+=(ye)+=y+=y by Lemma 2.2 and Proposition 2.4, and so ex=ey. This fact and its dual give that ϕ is bijection. We assert that piϕ,i=e=piϕ,jpjϕ,i for all i,jI. In fact, take i=xe,j=yeI where x,yPT. Then iϕ=ex and jϕ=ey. This implies by Lemma 2.2 and Proposition 2.4 that piϕ,i=(iϕ)i=(ex)(xe)=exe=(ex)+=e+=e and

    piϕ,jpjϕ,i=((iϕ)j)((jϕ)i)=exyeeyxe=exyeyxe=(e(xyeyx))+=e+=e.

    On the other hand, for every tT, we have (t+e)+=t+ and (et)=t by Proposition 2.4. In view of Lemma 2.2 (4), I and Λ can index the non-zero ˜RU-classes and ˜LU-classes of T0, respectively. Denote M={aTa+=a=e}{0}. For iI and λΛ, let ri=i and qλ=λ and denote pλi=qλri=λi. Since (t+e)+=t+, (et)=t, (ete)+=(ete)=e and t=(t+e)ete(et) by Lemma 2.2 and Proposition 2.4, in view of the proof of Theorem 3.6 in Lawson [14],

    θ:T0S=M0(I,Λ,M,P),t(t+e,ete,et),00

    is a semigroup isomorphism. Moreover, if we define

    (i,a,λ)+=(i,e,iϕ),(i,a,λ)=(λϕ1,e,λ),0+=0=0

    on S, then we can see that θ also preserves + and by Lemma 2.2 and Proposition 2.4. By the construction of P=(pλi), θ|T is a (2,1,1)-isomorphism from T onto M(I,Λ,M,P).

    In Theorem 3.1, if we identify i with iϕ for all iI, we can assume that I=Λ. So we have the following corollary.

    Corollary 3.2. Let I be a set and M a monoid with |I×M|1. Assume that P=(pλi)I×I is an I×I-matrix over M satisfying pii=e=pijpj,i for all i,jI, where e is the identity of M. Define a binary and two unary operations on the set

    S=M(I,M,P)={(i,x,j)i,jI,xM}

    as follows:

    (i,x,j)(k,y,l)=(i,xpjky,l),(i,x,j)+=(i,e,i),(i,x,j)=(j,e,j).

    Then (S,,+,) is a projection-primitive P-Ehresmann semigroup without zero or with zero which is not a projection. Conversely, every such semigroup can be obtained in this way.

    In this section, we consider the structures of projection-primitive P-Ehresmann semigroups with zero as a projection. To this aim, we need to recall some necessary notions and results. From Jones [7], a left projection algebra consists of a nonempty set P and a binary operation "×" satisfying the following axioms:

    (P1) e×e=e.

    (P2) e×(e×f)=(e×f)×e=e×f.

    (P3) (e×f)×g=e×(f×(e×g)).

    (P4) e×(f×g)=(e×f)×(e×(f×g)).

    For simplicity, we use the words "projection algebra" to replace "left projection algebra" in the sequel. Let (P,×) be a projection algebra. Define a relation "P" on P by the rule that for all e,fP, ePfifandonlyife=f×e. Then P is a partial order on P by (P1)–(P4). Moreover, by (P2) it is easy to see that

    e×fPe (4.1)

    for all e,fP. A projection algebra (P,×) with the least element 0 with respect to P is called primitive if no two different elements in P{0} can be compatible. On primitive projection algebras, we have the following simple results.

    Proposition 4.1. A primitive projection algebra (P,×) is just a (2, 0)-type algebra (P,×,0) satisfying the following conditions: For all e,fP,

    (Pr1) e×e=e.

    (Pr2) 0×e=0=e×0.

    (Pr3) e×f=0ore×f=e.

    (Pr4) e×f=0ifandonlyiff×e=0.

    In particular, if e×f=f×e for all e,fP, then e×f0 if and only if e=f0.

    Proof. Let (P,×) be a primitive projection algebra with the least element 0. We only need to show that (Pr2)–(Pr4) hold. Let e,fP. Since 0Pe, we have e×0=0, and so 0×e=(e×0)×e=e×0=0 by (P2). This proves (Pr2). In view of (4.1), (Pr3) is true. Finally, if e×f=0 and f×e0, then f×e=f0 by (Pr3). However,

    f×e=(f×e)×e=f×(e×(f×e))=f×(e×f)=f×0=0

    by (P3) and (Pr2). This is a contradiction. So (Pr4) holds.

    Conversely, let (P,×,0) be a (2, 0)-type algebra satisfying the given conditions. We only need to show that (P2)–(P4) hold. Let e,f,gP. By (Pr3), e×f=0 or e×f=e. In the former case, all items in (P2) are equal to 0 by (Pr2). In the latter case, all items in (P2) are equal to e by (Pr1). This shows (P2). Moreover, we also have e×g=0 or e×g=e. Then the following four cases may occur:

    (1)e×f=e,e×g=e;(2)e×f=e,e×g=0;(3)e×f=0,e×g=e;(4)e×f=0,e×g=0.

    In case (1), (e×f)×g=e×g=e and e×(f×(e×g))=e×(f×e). By (Pr3) and (Pr4), f×e=f in the case. So e×(f×e)=e×f=e. This proves (P3) for case (1). The other cases can be showed similarly. Finally, we consider (P4). By (Pr3), f×g=f or f×g=0. In the former case, the left side of (P4) is e×f, the right side of (P4) is (e×f)×(e×f)=e×f by (Pr1), and so they are equal. In the latter case, the two sides of (P4) are both 0 by (Pr2). The final result of the proposition follows from (Pr1), (Pr3) and (Pr4).

    By the dual of Proposition 2.4 in [7], and Propositions 2.7 and 4.1, we have the following result easily.

    Lemma 4.2. Let (S,,+,) be a primitive P-Ehresmann semigroup with zero 0 and 0PS. Define a binary operation "×S" on PS as follows: For all e,fPS, e×Sf=(ef)+=efe. Then (PS,×S,0) forms a primitive projection algebra. In particular, if S is Ehresmann, then e×Sf=f×Se for all e,fPS by Lemma 2.1.

    Let C be a nonempty set with a partial binary operation and (P,×,0) a primitive projection algebra with (P{0})C. Assume that d:CP,xd(x),  r:CP,xr(x) are maps such that d(C)r(C)(P{0}) and

    d(e)=e=r(e) (4.2)

    for all e(P{0}). According to Wang [26], C=(C,,d,r,P) is called a generalized category over (P,×,0) if the following conditions hold:

    (G1) For all x,yC, xy is defined if and only if r(x)×d(y)0 and then d(xy)=d(x) and r(xy)=r(y).

    (G2) If x,y,zC such that both xy and yz are defined, then (xy)z=x(yz).

    (G3) For all xC, d(x)x and xr(x) are defined and d(x)x=x=xr(x).

    (G4) If e,fP and e×f0, then (ef)e=e.

    If e×f=f×e for all e,fP, then e×f0 if and only if e=f, and so (G4) is always satisfied by (4.2), (G3) and Proposition 4.1. In this case, C=(C,,d,r,P) is a category in usual sense.

    Proposition 4.3. Let C=(C,,d,r,P) be a generalized category over the primitive projection algebra (P,×,0). Put C0=C{0}. Define a binary operation on C0 as follows: If x,yC and xy is defined in C, then xy=xy; all other products in C0 are 0. Moreover, define two unary operations on C0 as follows: 0=0=0 and x=d(x),x=r(x) for all xC. With these operations C0 is a projection-primitive P-Ehresmann semigroup with 0 as a projection. In the sequel, we call (C0,,,) a generalized category with zero adjoined.

    Proof. Let x,y,zC0. It is routine to check that (xy)z=0 precisely when x(yz)=0. Thus C0 is a semigroup by (G2). We shall show that the identities (ⅰ)–(ⅴ) and (ⅰ)–(ⅴ) are satisfied. Let x,yC0. If 0{x,y}, the identities (ⅰ)–(ⅴ) and (ⅰ)–(ⅴ) are satisfied obviously. So we assume that x,yC. Firstly, since d(x)x=x by (G3), we have xx=x. This gives (ⅰ). Dully, we have (ⅰ). Secondly, since d(y)=d(d(y))=d(y) by (4.2), it follows that xy0 if and only if xy0 by (G1). If this is the case, we have

    (xy)=d(xy)=d(x)=d(xy)=(xy)

    by (G1) again. This is exactly the identity (ⅱ). Dually, (ⅱ) is also true. Thirdly, by (4.2), (G1) and (Pr4), we can see that

    xy0d(x)×d(y)0xyx0.

    In this case, we have

    (xy)=d(d(x)d(y))=d(d(x))=d(x)

    by (G1) and xyx=d(x)d(y)d(x)=d(x) by (G4). This implies that the identity (ⅲ) is true. Dually, (ⅲ) is valid. Moreover, by (4.2), (G1) and (G3) we have

    xx=d(x)d(x)=d(d(x))d(x)=d(x).

    This gives (ⅳ). Similarly, we have (ⅳ). The identities (ⅴ) and (ⅴ) follow from the fact (x)=r(d(x))=d(x)=x by (4.2) and its dual. We have shown that (C0,,,) is a P-Ehresmann semigroup with the set of projections

    PC0={xxC0}=P={d(x)xC}{0}={r(x)xC}{0}.

    Finally, let e,fPC0 and ef. Then e=ef=fe. If e0, then fe is defined and so e=d(e)=d(fe)=d(f)=f by (4.2) and (G1). Thus, (C0,,,) is projection-primitive.

    Remark 4.4. Let C=(C,,d,r,P) be a generalized category over the primitive projection algebra (P,×,0). If P contains at least three elements and e×f0 for all e,fP{0}, then xy is defined for all x,yC. By Propositions 4.3 and 2.5, (C,,,) is a projection-primitive P-Ehresmann semigroup without zero. On the other hand, if P contains two elements, say, P={0,1}, then (C,,,) is a reduced P-Ehresmann semigroup. In fact, (C,) is a monoid with 1 as identity, and (C0,) is a monoid with zero adjoined. Thus we can think that generalized categories with zero adjoined covers the semigroups considered in the last section.

    Now we can give the main result of this section, which is a generalization of a result on restriction semigroups obtained by Jones in Section 4 of [10].

    Theorem 4.5. Let (S,,+,) be a P-Ehresmann semigroup with zero as a projection and |S|>1. Then S is projection-primitive if and only if S is (2, 1, 1)-isomorphic to a generalized category with zero adjoined.

    Proof. We have proved in Proposition 4.3 that every generalized category with zero adjoined is a projection-primitive P-Ehresmann semigroup with zero as a projection. To prove the converse, let (S,,+,) be a projection-primitive P-Ehresmann semigroup with zero 0 and 0PS. By Lemma 4.2, (PS,×S,0) forms a primitive projection algebra. Denote C=S{0}. Define a partial binary operation "" as follows:

    xy={xyif xy0,undefinedif xy=0, (4.3)

    where xy denotes the multiplication of x and y in the semigroup S. Define maps

    d:CPS,xx+,  r:CPS,xx. (4.4)

    Then we have d(C)r(C)(PS{0}) by Proposition 2.7 (1), and d(e)=e=r(e) for all ePS{0} by Lemma 2.2, respectively.

    We assert that (C,,d,r,PS) is a generalized category over the primitive projection algebra (PS,×S,0). First, let x,yC. By Proposition 2.7 (2) and (Pr3), (Pr4),

    xyisdefinedxy+x=x
    y+xy+=y+r(x)×Sd(y)0d(y)×Sr(x)0.

    In this case, by the identities defining P-Ehresmann semigroups and Lemma 2.2 we have

    d(xy)=d(xy)=(xy)+=(xy+)+=(xxy+)+
    =(x(xy+)+)+=(xxy+x)+=(xx)+=x+=d(x).

    Dually, we have r(xy)=r(y). Thus (G1) holds. Next, let x,y,zC, and xy and yz be defined. Since r(xy)=r(y) and d(yz)=d(y), (xy)z and x(yz) are defined, and so (xy)z=(xy)z=x(yz)=x(yz). This gives (G2). Moreover, for xC, since x+x=x0, it follows that x+x is defined and d(x)x=x. Dually, xx is defined and xr(x)=x. Thus (G3) is true. Finally, let e,fPS and e×Sf0. Then f×Se0 by (Pr4) and e=e×Sf=efe by (Pr3). In view of (G2), (ef)e is defined and (ef)e=(ef)e=efe=e. Thus (G4) is valid.

    By Proposition 4.3, we have a generalized category with zero adjoined (C0,,,). We shall show that S is (2, 1, 1)-isomorphic to C0. Define a map ψ:SC0 by assigning 0ψ=0 and xψ=x for all xC=S{0}. Obviously, ψ is a bijection. Let x,yS. If x=0 or y=0, then xψ=0 or yψ=0, whence (xy)ψ=0ψ=0=(xψ)(yψ). Assume that x,yC=S{0}. Then xy=0 in (S,,+,) if and only if xy is not defined in the generalized category (C,,d,r,PS), if and only if xy=0 in (C0,,,) by (4.3) and Proposition 4.3. This implies that (xy)ψ=(xψ)(yψ) for all x,yC. Thus ψ is a semigroup homomorphism. Furthermore, observe that 0+=0 by Proposition 2.7 (1) and 0=0 by Proposition 4.3, it follows that (0ψ)=0=0=0ψ=0+ψ. Dually, we have (0ψ)=0ψ. If xC=S{0}, then x+C by Proposition 2.7 (1), this implies that (xψ)=x=d(x)=x+=x+ψ by Propositions 4.3 and (4.4). Dually, we have (xψ)=xψ for all xC. We have shown that ψ is a (2, 1, 1)-isomorphism. By Lemma 4.2 and the statements before Proposition 4.3, we have the following result appeared in Jones [10].

    Corollary 4.6 ([10]). Let (S,,+,) be an Ehresmann semigroup with zero as a projection and |S|>1. Then S is projection-primitive if and only if S is (2, 1, 1)-isomorphic to a category with zero adjoined.

    Remark 4.7. Let (S,,+,) be a primitive P-Ehresmann semigroup without zero or with zero 0 but 0PS. Let S and define additionally x=x== and +==. Then (S,,+,) forms a primitive P-Ehresmann semigroup with zero and PS. In this case, the generalized category associated with S constructed in the proof of Theorem 4.5 is just (S,,+,) and the corresponding generalized category with zero adjoined is just (S,,+,). By Remark 4.4, we can think that Theorem 4.5 also works for the semigroups considered in the last section. However, it is trivial certainly in the case.

    In this paper, we have obtained the structures of projection-primitive P-Ehresmann semigroups. As a future work, one can investigate the associative algebras of these semigroups by using the results obtained in the present paper.

    The author expresses his profound gratitude to the referees for the valuable comments and suggestions, which improve greatly the content and presentation of this article. In particular, according to the advices of one of the referees, we rewrite Section 3 (with the help of the results of Lawson [14] provided by the referee) and Section 4. This research is supported by the National Natural Science Foundation of China (11661082). Thanks also go to the editor for the timely communications.

    The author declares there is no conflict of interest.



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