The Research On Spectral Duality Theory Of Posets And Semigroups

In 1937,Stone established the dually equivalence of the category of Boolean al-gebras to certain category of topological spaces.Later,Stone generalized the result for Boolean algebras to distributive lattices and showed that the category of distributive lattices and lattice homomorphisms is dually equivalent to that of spectral spaces and spectral maps.Following Stone's ideas,many scholars have established lots of dualities for various order and algebra structures,such as rings,distributive meet-semilattices,semilattices with unit,distributive join-semilattices,bounded lattices,distributive poset-s,etc.Recently,the spectra of various structures has attracted the attention of many scholars.In this thesis,we study the spectra on posets,semilattices,lattices,semigroups and ordered semigroups,and obtain the results in the following chapters.In Chapter 3,we discuss the poset P without a least element.The topological space we focus on is the set of all prime semi-ideals of P endowed with hull-kernel topology.We observe that the family of all nonempty super-compact open subsets in this space forms a basis for the topology and corresponds to the elements in P one by one.This discovery inspires us to define a new topological space,PP*-space,which is exactly the dual space of a poset without a least element.We build a category PS whose objects are posets without a least element and arrows are order-preserving maps satisfying that the inverse image of a prime semi-ideal is prime.And we show that PS and PP*T are dually categorically equivalent,where PP*T has PP*-spaces as objects and maps,which satisfy the inverse image of a nonempty super-compact open set is nonempty super-compact open,as arrows.Similarly,we can get topological dualities for posets with a least element,semilattices,and lattices,respectively.Moreover,we show that the dual spaces of posets without a least element are weak sober but not always sober,and the dual spaces of posets with a least element are sober.Particularly,the dual spaces of semilattices without a least element are weak sober but not sober.In Chapter 4,we aim to investigate the Zariski topology on prime ideals in a com-mutative semigroup S,denoted by Spec(S).First,we show that a topological space X is homeomorphic to Spec(S)for some commutative semigroup S with 0 if and only if X is an SP-space that can be described purely in topological terms;X is homeomorphic to Spec(S)for some commutative semigroup S without 0 if and only if X is an SP*-space.Next,we prove that an adjunction exists between the category of commutative semigroups with 0 and that of SP-spaces;an adjunction exists between the category of commutative semigroups without 0 and that of SP*-spaces.In Chapter 5,we not only develops the theory of prime ideals but also investigates lots of topological properties on the space of prime ideals in ordered semigroups.At first,to ensure the existence of prime ideals,a class of ordered semigroups which will be denoted by SIP are studied,and we show that if S is a commutative or intra-regular ordered semigroup,then S ? SIP.And then,the hull-kernel topology for prime ideals Spec(S)is defined and some topological properties like separation axioms,compact-ness,connectedness are studied.Furthermore,the subspace M(S,I),whose elements are the minimal prime ideals containing the ideal I in an ordered semigroup is investi-gated.Moreover,if S belongs to SIP,then M(S,I)is Hausdorff,totally disconnected,completely regular and has a base L(S,I)consisting of clopen subsets.Finally,a result is obtained that if S is a pseudocomplemented commutative ordered semigroup,then the minimal prime ideal space M(S)is a Stone space.In Chapter 6,our main purpose is to develop the spectral theory of prime L-ideals in ordered semigroups.First,the concept of an L-fuzzy ideal and that of an L-fuzzy prime ideal are given and discussed.And then a topological space,called the L-fuzzy prime spectrum of an ordered semigroup,has been gained.This topological space is T0 and sober.Furthermore some homeomorphic spaces of this topological space are investigated.Finally,a contravariant functor from the category of commutative ordered semigroups into the category of sober spaces is obtained.