C*-Algebras Generated by Weakly Quasi-Lattice Ordered Groups

Illinois journal of mathematics

Semigroup Forum, 1985

Journal of Physics: Conference Series, 2019

LetGbe a group equipped with a cyclic order such thatGbe a cyclically ordered group. Under the given order and a particular condition onG, the positive coneP(G) ofGis a cyclically ordered semigroup. We induced the representation of aC*-algebraAin a Hilbert space such that we obtain a covariant representation of an action ofP(G) by automorphism ofA. C*-dynamical system (A,α,P(G)), whereαis an action ofP(G) by automorphism ofA.

Czechoslovak Mathematical Journal, 1999

Journal of the Australian Mathematical Society, 2003

Let A be a uniformly complete vector sublattice of an Archimedean semiprime f-algebra B and p ∈ {1, 2,…}. It is shown that the set ΠBp (A) = {f1 … fp: fk ∈ A, k = 1, …, p } is a uniformly complete vector sublattice of B. Moreover, if A is provided with an almost f-algebra multiplication * then there exists a positive operator Tp, from ΠBp(A) into A such that fi *…* fp = Tp(f1 …fp) for all f1…fp ∈ A.As application, being given a uniformly complete almost f-algebra (A, *) and a natural number p ≧ 3, the set Π*p(A) = {f1 *… *fp: fk ∈ A, k = 1…p} is a uniformly complete semiprime f-algebra under the ordering and the multiplication inherited from A.

In this paper, the notion of an ordered Γ-semigroup is introduced and some examples are given. Further the terms commutative ordered Γ-semigroup, quasi commutative ordered Γ-semigroup, normal ordered Γsemigroup, left pseudo commutative ordered Γsemigroup, right pseudo commutative ordered Γsemigroup are introduced. It is proved that (1) if S is a commutative ordered Γ-semigroup then S is a quasi commutative ordered Γ-semigroup, (2) if S is a quasi commutative ordered Γ-semigroup then S is a normal ordered Γ-semigroup, (3) if S is a commutative ordered Γ-semigroup, then S is both a left pseudo commutative and a right pseudo commutative ordered Γ-semigroup. Further the terms; left identity, right identity, identity, left zero, right zero, zero of an ordered Γ-semigroup are introduced. It is proved that if a is a left identity and b is a right identity of an ordered Γ-semigroup S, then a = b. It is also proved that any ordered Γsemigroup S has at most one identity. It is proved that if a is a left zero and b is a right zero of an ordered Γsemigroup S, then a = b and it is also proved that any ordered Γ-semigroup S has at most one zero element. The terms; ordered Γ-subsemigroup, ordered Γsubsemigroup generated by a subset, α-idempotent, Γ-idempotent, strongly idempotent, midunit, r-element, regular element, left regular element, right regular element, completely regular element, (α, β)-inverse of an element, semisimple element and intra regular element in an ordered Γ-semigroup are introduced. Further the terms idempotent ordered Γ-semigroup and generalized commutative ordered Γ-semigroup are introduced. It is proved that every α-idempotent element of an ordered Γ-semigroup is regular. It is also proved that, in an ordered Γ-semigroup, a is a regular element if and only if a has an ( , )-inverse. It is proved that, (1) if a is a completely regular element of an ordered Γ-semigroup S, then a is both left regular and right regular, (2) if "a" is a completely regular element of an ordered Γ-semigroup S, then a is regular and semisimple, (3) if "a" is a left regular element of an ordered Γ-semigroup S, then a is semisimple, (4) if "a" is a right regular element of an ordered Γ-semigroup S, then a is semisimple, (5) if "a" is a regular element of an ordered Γsemigroup S, then a is semisimple and (6) if "a" is a intra regular element of an ordered Γsemigroup S, then a is semisimple. The term separative ordered Γ-semigroup is introduced and it is proved that, in a separative ordered Γ-semigroup S, for any x, y, a, b ∈ S, the statements (i) xΓa ≤ xΓb if and only if aΓx ≤ bΓx, (ii) xΓ xΓa ≤ xΓxΓb implies xΓa ≤ xΓb,(iii) xΓyΓa ≤ xΓyΓb implies yΓxΓa ≤ yΓxΓb hold.

Journal of Algebra, 2009

In this article we give some necessary and sufficient conditions for a lattice-ordered semigroup algebra to be isomorphic to a latticeordered triangular matrix algebra.

Glasgow Mathematical Journal, 2008

In this paper we introduce the notion of normally ordered block-group as a natural extension of the notion of normally ordered inverse semigroup considered previously by the author. We prove that the class NOS of all normally ordered blockgroups forms a pseudovariety of semigroups and, by using the Munn representation of a block-group, we deduce the decompositions in Mal'cev products NOS = EI m POI and NOS ∩ A = N m POI, where A, EI and N denote the pseudovarieties of all aperiodic semigroups, all semigroups with just one idempotent and all nilpotent semigroups, respectively, and POI denotes the pseudovariety of semigroups generated all semigroups of injective order-preserving partial transformations on a finite chain. These relations are obtained after showing that BG = EI m Ecom = N m Ecom, where BG and Ecom denote the pseudovarieties of all block-groups and all semigroups with commuting idempotents, respectively.

CGASA, Vol. 13, NO. 1, 2020

The category, or class of algebras, in the title is denoted by W. A hull operator (ho) in W is a reflection in the category consisting of W objects with only essential embeddings as morphisms. The proper class of all of these is hoW. The bounded monocoreflection in W is denoted B. We classify the ho's by their interaction with B as follows. A "word" is a function w : hoW −→ W W obtained as a finite composition of B and x a variable ranging in hoW. The set of these,"Word", is in a natural way a partially ordered semigroup of size 6, order isomorphic to F(2), the free 0 − 1 distributive lattice on 2 generators. Then, hoW is partitioned into 6 disjoint pieces, by equations and inequations in words, and each piece is represented by a characteristic order-preserving quotient of Word (≈ F(2)). Of the 6: 1 is of size ≥ 2, 1 is at least infinite, 2 are each proper classes, and of these 4, all quotients are chains; another 1 is a proper class with unknown quotients; the remaining 1 is not known to be nonempty and its quotients would not be chains.

Algebra Universalis, 1991

Let R(n, u, I) denote the class of all algebras isomorphic to ones whose elements are binary relations and whose operations are union, intersection, and relation composition (or relative product) of relations. We prove that R(w, c~, l) is not a variety and is not finitely axiomatizable. Let DLOS denote the class of all structures (A, v, ^, o) where (A, v, ^ ) is a distributive lattice, (A, o) is a semigroup and o is additive w.r.t.v. We prove that DLOS is the variety generated by R(u, n, [), and moreover, if (A, v, A, o) e DLOS then it is representable whenever we disregard one of its operations.