L-slope Algebra

Incline theory originates from control theory, briefly, an incline algebra is an additive idempotent semiring in which the multiplication of any two elements are less than or equal to each factor. Many well known algebras such as boolean algebra, distributive lattice are all examples of incline. Incline and incline matrix theory have very good prospect in automata theory, decision-making theory, control theory, graph theory, nervous system, and so on.In this paper, the order aspect and categorical properties of inclines are disscussed, the main contents are as follows:(1) The concepts of L-subinclines, L-ideals, L-filters and L-congruence relations of inclines are defined, and some characterizations of them are given.(2) Operations of L-subinclines, L-ideals, L-filters and L-congruence relations of inclines are studied. Properties of L-subinclines, L-ideals, L-filters and L-congruence relations of inclines under incline homomorphism are disscussed. It is proved that if L is a frame, L-subincline is preserved by incline homomorphisms, L-filter is preserved by incline epimorphisms, and some examples are given to show that L-ideals and L-congruence relations are not preserved by incline epimorphism even if L is a frame. The intersection (∧) and union (∨) operations of L-subinclines (resp.,L-ideals, L-filters, L-congruence relations) are defined, and it is proved that the intersection of a class of L-subinclines (resp.,L-ideals, L-filters, L-congruence relations) of an incline (X,+,*) is also an L-subincline (resp.,L-ideal, L-filter, L-congruence relation) of (X,+,*), if L is a∧-continous lattice, the union of a directed class of L-subinclines (resp.,L-ideals, L-filters, L-congruence relations) of (X,+,*) is also an L-subincline (resp.,L-ideal, L-filter, L-congruence relation) of (X,+,*). The direct product and direct sum of L-subinclines (resp.,L-ideals, L-filters) are also defined, and it is proved that the direct product and direct sum of a class of L-subinclines (resp.,L-ideals, L-filters) of an incline (X,+,*) is also an L-subincline (resp.,L-ideal, L-filter) of (X,+,*). The direct product of L-congruence relations of an incline (X,+,*) is also defined, and it is proved that the direct product of a class of L-congruence relations of (X,+,*) is also an L-congruence relation of (X,+,*). The quotient of an incline and an L-subincline (resp.,L-ideal, L-filter) are defined, and it is proved that the quotient of an L-subincline (resp., L-filter) of (X,+,*) is an L-subincline(resp.,L-filter)of the quotient algebra of(X,+,*).(3) Order aspect of L-subinclines,L-ideals,L-filters and L-congruence relations of inclines are studied.It is proved that(SIL(X,+,*),≤)(the set of all L-subinclines of an incline(X,+,*)),(IdL(X,+,*),≤)(the set of all L-ideals of an incline(X,+,*)),and (FilL(X,+,*),≤)(the set of all L-filters of an incline(X,+,*))are all∧-continuous com-plete sublattices of(LX,≤),and(EquL(X,+,*),≤)(the set of all L-equivalence relations on an incline(X,+,*))and(ConL(X,+,*),≤)(the set of all L-congruence relations on an incline(X,+,*))are both∧-continuous complete sublattices of(LX×X,≤),and the formation of the infimum and supremum of a class of elements in(SIL(X,+,*),≤) (resp.,(IdL(X,+,*),≤),(FilL(X,+,*),≤),(EquL(X,+,*),≤)),the supremum of a class of elements in(ConL(X,+,*),≤)are given.(4)Relations between L-ideals and L-congruence relations,L-filters and L-congruence relations are studied. The concepts of regular L-congruence relation,and normal L-congruence relation are defined. And it is proved that,if L is a frame,for an incline (X,+,*)with a zero element 0,(IdL(X,+,*),≤)is lattice isomorphic to(RConL(X,+,*),≤)(the set of all regular L-congruence relations on(X,+,*)),and for an incline(X,+,*) with an identity 1,there exists a one-to-one correspondence between FilL(X,+,*)and NConL(X,+,*)(the set of all normal L-congruence relations on(X,+,*)).(5) Categorical aspects of L-subinclines,L-ieals and L-filters are studied. Cate-gories L-SInc,L-Id,L-Fil are defined,and L-Id,L-Fil are proved to be full reflictive subcategories of L-SInc,the formation of equalizer and product in L-SInc,L-Id,L-Fil are given.L-SInc,L-Id,L-Fil are all proved to be topological categories on Inc and have pull back.L-cotower of inclines,L-cotower of incline ideals and L-cotower of fil-ters are defined,and the category IncLC consisting of all L-cotowers of inclines,category IncLC│Id consisting of all L-cotowers of incline ideals,category IncLC│Fil consisting of all L-cotowers of incline filters are also defined. It is proved that,if L is frame and satis-fies∨{b∈L│b