An element of a semigroup S is called irreducible if it cannot be expressedas a product of two elements in S both distinct from itself. In this paper we showthat the class C of all completely regular monoids with irre...An element of a semigroup S is called irreducible if it cannot be expressedas a product of two elements in S both distinct from itself. In this paper we showthat the class C of all completely regular monoids with irreducible identity elementssatises the strong isomorphism property and so it is globally determined.展开更多
Congruence is a very important aspect in the study of the semigroup theory.In general,the Kernel-trace characterizations,Green's relations and subvarieties are main tools in the consideration of congruences on com...Congruence is a very important aspect in the study of the semigroup theory.In general,the Kernel-trace characterizations,Green's relations and subvarieties are main tools in the consideration of congruences on completely regular semigroups.In this paper,we give one class of congruences on completely regular semigroups with the representation of wreath product of translational hulls on completely simple semigroups.By this new way,the least Clifford semigroup congruences on completely regular semigroups are generalized.展开更多
In this paper, we give some characterizations of almost completely regular spaces and c-semistratifiable spaces（CSS） by semi-continuous functions. We mainly show that：（1）Let X be a space. Then the following state...In this paper, we give some characterizations of almost completely regular spaces and c-semistratifiable spaces（CSS） by semi-continuous functions. We mainly show that：（1）Let X be a space. Then the following statements are equivalent：（i） X is almost completely regular.（ii） Every two disjoint subsets of X, one of which is compact and the other is regular closed, are completely separated.（iii） If g, h ： X → I, g is compact-like, h is normal lower semicontinuous, and g ≤ h, then there exists a continuous function f ： X → I such that g ≤ f ≤ h;and（2） Let X be a space. Then the following statements are equivalent：（a） X is CSS;（b） There is an operator U assigning to a decreasing sequence of compact sets（Fj）j∈N,a decreasing sequence of open sets（U（n,（Fj）））n∈N such that（b1） Fn■U（n,（Fj）） for each n ∈ N;（b2）∩n∈NU（n,（Fj）） =∩n∈NFn;（b3） Given two decreasing sequences of compact sets（Fj）j∈N and（Ej）j∈N such that Fn■Enfor each n ∈ N, then U（n,（Fj））■U（n,（Ej）） for each n ∈ N;（c） There is an operator Φ ： LCL（X, I） → USC（X, I） such that, for any h ∈ LCL（X, I）,0 Φ（h） h, and 0 〈 Φ（h）（x） 〈 h（x） whenever h（x） 〉 0.展开更多
Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The gen...Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The general procedure is as follows: A suitable equivalence relation is defined on an enlargement <sup>*</sup>X of the space X which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group. Accordingly, every f in C(X,R) (the space of bounded continuous real valued functions on X) or Cc(X,R) (the space of continuous real valued functions on X with compact support) or the dual group <span style="white-space:nowrap;">Γ of the locally compact Abelian group G is extended to the set <img alt="" src="Edit_b9535172-924d-44f0-bab3-c49db17a3b7a.png" /> of the above mentioned equivalence classes. A compact topology on <img alt="" src="Edit_9d7962a3-b8a3-4693-b62a-078c8c4b4853.png" /> is obtained as the weak topology generated by these extensions of f. Then X is naturally imbedded densely in <img alt="" src="Edit_f7d403b2-eff3-4555-b8e7-1b106e06d2e7.png" />.展开更多
The enhanced power graph P_(E)(S)of a semigroup S is a simple graph whose vertex set is S and two vertices a,y∈S are adjacent if and only if c,y∈(z)for some z∈S,where(z)is the subsemigroup generated by z.In this pa...The enhanced power graph P_(E)(S)of a semigroup S is a simple graph whose vertex set is S and two vertices a,y∈S are adjacent if and only if c,y∈(z)for some z∈S,where(z)is the subsemigroup generated by z.In this paper,we first describe the structure of P_(E)(S)for an arbitrary semigroup S,and then discuss the connectedness of P_(E)(S).Further,we characterize the semigroup S in the cases when P_(E)(S)is separately a complete,bipartite,regular,tree and null graph.The planarity,together with the minimum degree and independence number,of P_(E)(S)is also investigated.The chromatic number of a spanning subgraph,i.e.,the cyclic graph,of P_(E)(S)is proved to be countable.In the final part of this paper,we construct an example of a semigroup S such that the chromatic number of P_(E)(S)need not be countable.展开更多
Let G be a transitive group.We mainly prove that the group(G,a)is primitive for a given special permutation a and the semigroup(G,a)is synchronizing for a given special transformationα.
The problem as to whether the sub-T0 separation and complete regularity are invariant under homeomorphism is answered negatively.And,the problem of multiplicativity of the complete regularity in general L-fuzzy topolo...The problem as to whether the sub-T0 separation and complete regularity are invariant under homeomorphism is answered negatively.And,the problem of multiplicativity of the complete regularity in general L-fuzzy topological spaces is also answered negatively.展开更多
A structure theorem for superabundant semigroups in terms of semilattices of normalized Rees matrix semigroups over some cancellative monoids is obtained. This result not only provides a construction method for supera...A structure theorem for superabundant semigroups in terms of semilattices of normalized Rees matrix semigroups over some cancellative monoids is obtained. This result not only provides a construction method for superabundant semigroups but also generalizes the well-known result of Petrich on completely regular semigroups. Some results obtained by Fountain on abundant semigroups are also extended and strengthened.展开更多
The aim of this paper is to study regular orthocryptogroups. After obtaining some charac- terizations of such semigroups, we establish the construction theorem of regular orthocryptogroups. As an application, we give ...The aim of this paper is to study regular orthocryptogroups. After obtaining some charac- terizations of such semigroups, we establish the construction theorem of regular orthocryptogroups. As an application, we give the construction theorem of right quasi-normal orthocryptogroups and study homomorphisms between two regular orthocryptogroups.展开更多
基金The NSF(11261021) of Chinathe NSF(20142BAB201002) of Jiangxi Province
文摘An element of a semigroup S is called irreducible if it cannot be expressedas a product of two elements in S both distinct from itself. In this paper we showthat the class C of all completely regular monoids with irreducible identity elementssatises the strong isomorphism property and so it is globally determined.
基金National Natural Science Foundation of China(No.11671056)General Science Foundation of Shanghai Normal University,China(No.KF201840)。
文摘Congruence is a very important aspect in the study of the semigroup theory.In general,the Kernel-trace characterizations,Green's relations and subvarieties are main tools in the consideration of congruences on completely regular semigroups.In this paper,we give one class of congruences on completely regular semigroups with the representation of wreath product of translational hulls on completely simple semigroups.By this new way,the least Clifford semigroup congruences on completely regular semigroups are generalized.
基金Supported by the Project of Young Creative Talents of Guangdong Province(Grant No.2014KQNCX161)the Ph D Start-up Fund of Natural Science Foundation of Guangdong Province(Grant No.2014A030310187)+3 种基金the National Natural Science Foundation of China(Grant Nos.11526158613792111471153)the Research Fund for Higher Education of Fujian Province of China(Grant No.2013J01029)
文摘In this paper, we give some characterizations of almost completely regular spaces and c-semistratifiable spaces（CSS） by semi-continuous functions. We mainly show that：（1）Let X be a space. Then the following statements are equivalent：（i） X is almost completely regular.（ii） Every two disjoint subsets of X, one of which is compact and the other is regular closed, are completely separated.（iii） If g, h ： X → I, g is compact-like, h is normal lower semicontinuous, and g ≤ h, then there exists a continuous function f ： X → I such that g ≤ f ≤ h;and（2） Let X be a space. Then the following statements are equivalent：（a） X is CSS;（b） There is an operator U assigning to a decreasing sequence of compact sets（Fj）j∈N,a decreasing sequence of open sets（U（n,（Fj）））n∈N such that（b1） Fn■U（n,（Fj）） for each n ∈ N;（b2）∩n∈NU（n,（Fj）） =∩n∈NFn;（b3） Given two decreasing sequences of compact sets（Fj）j∈N and（Ej）j∈N such that Fn■Enfor each n ∈ N, then U（n,（Fj））■U（n,（Ej）） for each n ∈ N;（c） There is an operator Φ ： LCL（X, I） → USC（X, I） such that, for any h ∈ LCL（X, I）,0 Φ（h） h, and 0 〈 Φ（h）（x） 〈 h（x） whenever h（x） 〉 0.
文摘Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The general procedure is as follows: A suitable equivalence relation is defined on an enlargement <sup>*</sup>X of the space X which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group. Accordingly, every f in C(X,R) (the space of bounded continuous real valued functions on X) or Cc(X,R) (the space of continuous real valued functions on X with compact support) or the dual group <span style="white-space:nowrap;">Γ of the locally compact Abelian group G is extended to the set <img alt="" src="Edit_b9535172-924d-44f0-bab3-c49db17a3b7a.png" /> of the above mentioned equivalence classes. A compact topology on <img alt="" src="Edit_9d7962a3-b8a3-4693-b62a-078c8c4b4853.png" /> is obtained as the weak topology generated by these extensions of f. Then X is naturally imbedded densely in <img alt="" src="Edit_f7d403b2-eff3-4555-b8e7-1b106e06d2e7.png" />.
基金the support of MATRICS Grant(MTR/2018/000779)funded by SERB,India.
文摘The enhanced power graph P_(E)(S)of a semigroup S is a simple graph whose vertex set is S and two vertices a,y∈S are adjacent if and only if c,y∈(z)for some z∈S,where(z)is the subsemigroup generated by z.In this paper,we first describe the structure of P_(E)(S)for an arbitrary semigroup S,and then discuss the connectedness of P_(E)(S).Further,we characterize the semigroup S in the cases when P_(E)(S)is separately a complete,bipartite,regular,tree and null graph.The planarity,together with the minimum degree and independence number,of P_(E)(S)is also investigated.The chromatic number of a spanning subgraph,i.e.,the cyclic graph,of P_(E)(S)is proved to be countable.In the final part of this paper,we construct an example of a semigroup S such that the chromatic number of P_(E)(S)need not be countable.
基金National Natural Science Foundation of China(12401024,11871360,12171222)Young Innovative Talents Project at Guangdong Universities(2021KQNCX048)+1 种基金Ph.D.Research Startup Foundation of Lingnan Normal University(ZL1924)Natural Science Foundation of Guangdong Province(2022A1515010964,2019A1515011031).
文摘Let G be a transitive group.We mainly prove that the group(G,a)is primitive for a given special permutation a and the semigroup(G,a)is synchronizing for a given special transformationα.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 19701020) the Teaching and Research Award for Outstanding Young Teachers in Higher Education Institutions of MOE, China.
文摘The problem as to whether the sub-T0 separation and complete regularity are invariant under homeomorphism is answered negatively.And,the problem of multiplicativity of the complete regularity in general L-fuzzy topological spaces is also answered negatively.
文摘A structure theorem for superabundant semigroups in terms of semilattices of normalized Rees matrix semigroups over some cancellative monoids is obtained. This result not only provides a construction method for superabundant semigroups but also generalizes the well-known result of Petrich on completely regular semigroups. Some results obtained by Fountain on abundant semigroups are also extended and strengthened.
基金The research is supported by NSF for youth of Shandong Province. China.
文摘The aim of this paper is to study regular orthocryptogroups. After obtaining some charac- terizations of such semigroups, we establish the construction theorem of regular orthocryptogroups. As an application, we give the construction theorem of right quasi-normal orthocryptogroups and study homomorphisms between two regular orthocryptogroups.
