Ideals in bounded equality algebras

Akbar Paad

doi:10.2298/fil1907113p

Ideals in bounded equality algebras

Filomat ◽

10.2298/fil1907113p ◽

2019 ◽

Vol 33 (7) ◽

pp. 2113-2123

Author(s):

Akbar Paad

Keyword(s):

Boolean Algebra ◽

Congruence Relation ◽

Isomorphism Theorem ◽

Equality Algebra ◽

One To One

In this paper, the concept of ideal in bounded equality algebras is introduced. With respect to this concepts, some related results are given. In particular, we prove that there is an one-to-one corresponding between congruence relation on an involutive equality algebra and the set of ideals on it. Also, we prove the first isomorphism theorem on equality algebras. Moreover, the notions of prime and Boolean ideals in equality algebras are introduced. Finally, we prove that ideal I of involutive prelinear equality algebra E is a Boolean ideal if and only if E/I is a Boolean algebra.

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Study of MV-algebras via derivations

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2019-0044 ◽

2019 ◽

Vol 27 (3) ◽

pp. 259-278

Author(s):

Jun Tao Wang ◽

Yan Hong She ◽

Ting Qian

Keyword(s):

Boolean Algebra ◽

Direct Product ◽

Galois Connection ◽

Product Representation ◽

Mv Algebra ◽

One To One ◽

The Relationship ◽

Mv Algebras ◽

Derivation Pair

AbstractThe main goal of this paper is to give some representations of MV-algebras in terms of derivations. In this paper, we investigate some properties of implicative and difference derivations and give their characterizations in MV-algebras. Then, we show that every Boolean algebra (idempotent MV-algebra) is isomorphic to the algebra of all implicative derivations and obtain that a direct product representation of MV-algebra by implicative derivations. Moreover, we prove that regular implicative and difference derivations on MV-algebras are in one to one correspondence and show that the relationship between the regular derivation pair (d, g) and the Galois connection, where d and g are regular difference and implicative derivation on L, respectively. Finally, we obtain that regular difference derivations coincide with direct product decompositions of MV-algebras.

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Normality and

Journal of Symbolic Logic ◽

10.2307/2275072 ◽

1991 ◽

Vol 56 (3) ◽

pp. 1064-1067

Author(s):

R. Zrotowski

Keyword(s):

Boolean Algebra ◽

Partial Answer ◽

Normal Ideal ◽

Mahlo Cardinal ◽

One To One ◽

Made In

AbstractThe main result of this paper is that if κ is not a weakly Mahlo cardinal, then the following two conditions are equivalent:1. is κ+-complete.2. is a prenormal ideal.Our result is a generalization of an announcement made in [Z]. We say that is selective iff for every -function f: κ → κ there is a set X ∈ such that f∣(κ − X) is one-to-one. Our theorem provides a positive partial answer to a question of B. Wȩglorz from [BTW, p. 90], viz.: is every selective ideal with κ+-complete, isomorphic to a normal ideal?The theorem is also true for fine ideals on [λ]<κ for any κ ≤ λ, i.e. if κ is not a weakly Mahlo cardinal then the Boolean algebra is λ+-complete iff is a prenormal ideal (in the sense of [λ/<κ).

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Computable Boolean algebras

Journal of Symbolic Logic ◽

10.2307/2695066 ◽

2000 ◽

Vol 65 (4) ◽

pp. 1605-1623 ◽

Cited By ~ 19

Author(s):

Julia F. Knight ◽

Michael Stob

Keyword(s):

Boolean Algebra ◽

Embedding Theorem ◽

Boolean Algebras ◽

Linear Ordering ◽

Isomorphism Theorem ◽

Embedding Theorems ◽

Trivial Case ◽

Linear Orderings ◽

Successor Relation

Feiner [F] showed that a Boolean algebra need not have a computable copy (see also [T2]). Downey and Jockusch [D-J] showed that every low Boolean algebra does have a computable copy. Thurber [T3], showed that every low2 Boolean algebra has a computable copy. Here we show that every Boolean algebra which is low3, or even low4, has a computable copy.The results of [D-J] and [T3] were obtained by passing to linear orderings. In [D-J], there is an embedding theorem saying that any linear ordering which is with the successor relation as an added predicate can be embedded in a slightly larger linear ordering which is computable. An isomorphism theorem of Remmel [R] is used to show that the interval algebras of the two linear orderings are isomorphic (except in a trivial case). In [T3], there is an embedding theorem saying that any linear ordering which is with certain added predicates can be embedded in one which is with successor. Again the isomorphism theorem of Remmel is used to show that the interval algebras are isomorphic (except in a trivial case).Here, instead of passing to linear orderings, we work directly with Boolean algebras. We begin with a review of the known results. We re-formulate the embedding theorems of Downey-Jockusch and Thurber in terms of Boolean algebras. We extract from Remmel's isomorphism theorem some information on complexity. In this way, we show that a low Boolean algebra is isomorphic to a computable one by an isomorphism which is , at worst, and the same is true for a low2 Boolean algebra.

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On the structure of Stone lattices

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700042076 ◽

1970 ◽

Vol 2 (3) ◽

pp. 401-413 ◽

Cited By ~ 1

Author(s):

P. D. Finch

Keyword(s):

Boolean Algebra ◽

Distributive Lattice ◽

Subdirect Product ◽

Stone Lattice ◽

One To One

C.C. Chen and G. Grätzer have shown that a Stone lattice is determined by a triple (C, D, ø) where C is a boolean algebra, D is a distributive lattice with 1 and ø is an e-homomorphism from C into D(D), the lattice of dual ideals of D.It is shown here that any Stone lattice is, up to an isomorphism, a subdirect product of its centre C(L) and a special Stone lattice M(L). Special Stone lattices are characterised, in the terminology of the Chen-Grätzer triple, by the fact that the e-homomorphism Φ is one to one.In this paper we characterise a special Stone lattice L as a triple (H, C, Do) where H is a distributive lattice with 0 and 1, C is a boolean e-subalgebra of the centre of H and Do is a sublattice of H with o such that d ∈ Do & c ∈ C = d ∧ c ∈ Do, and which separates the elements of C in the sense that for any c1 ≠c2 in C there is a d in Do with d ≤ c1 but d ≰ C2. It then turns out that C is C(L) and Do is the dual of D(L).

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A Support Program

Language Speech and Hearing Services in Schools ◽

10.1044/0161-1461.2502.112 ◽

1994 ◽

Vol 25 (2) ◽

pp. 112-114 ◽

Cited By ~ 3

Author(s):

Henna Grunblatt ◽

Lisa Daar

Keyword(s):

Hearing Aids ◽

School Counselor ◽

Support Program ◽

Educational Audiology ◽

Part Program ◽

One To One

A program for providing information to children who are deaf about their deafness and addressing common concerns about deafness is detailed. Developed by a school audiologist and the school counselor, this two-part program is geared for children from 3 years to 15 years of age. The first part is an educational audiology program consisting of varied informational classes conducted by the audiologist. Five topics are addressed in this part of the program, including basic audiology, hearing aids, FM systems, audiograms, and student concerns. The second part of the program consists of individualized counseling. This involves both one-to-one counseling sessions between a student and the school counselor, as well as conjoint sessions conducted—with the student’s permission—by both the audiologist and the school counselor.

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