Prime Ideal Theory for General Algebras

Erné, Marcel

doi:10.1007/978-94-017-2529-3_7

Cited by 7 publications

(10 citation statements)

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“…As in the theory of rings and other algebraic structures, also in order theory residuals [16,18] and polars (annihilators) [6,27,29,30,33] are useful tools. For subsets A and C of a poset Q , the set residual C : A is defined to be the set of…”

Section: Ideal Completions and Distributive Lawsmentioning

confidence: 99%

“…For posets instead of lattices, the discussion becomes more subtle, because there are various reasonable notions of ideals, of distributivity, of primeness, etc., and the absence of joins or meets may cause serious problems. This area of order theory was investigated in a series of papers by Erné [10][11][12][13][14][15][16][17][18][19][20] and independently by Chajda, Halas, Larmerová, Rachůnek, Niederle [5,6,26,27,29,30,36,39], and later by Joshi, Kharat, Mokbel, Mundlik, Waphare [31,32,34,49,50] and many others.…”

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confidence: 99%

“…The prime ideal theorems for Boolean algebras and distributive lattices (Stone [44,45]), for arbitrary lattices (Rav [40]), and for general algebras and posets (Erné [14,16,17]) all require choice principles equivalent to the Ultrafilter Principle, which postulates that every proper set-theoretical filter be contained in an ultrafilter-a principle not provable in choicefree set theory. As a ''constructive'' substitute, the presence of enough atoms or coatoms (dual atoms) allows to work in an entirely choice-free setting, because then weak distributivity assumptions give rise to certain prime filters or ideals, respectively.…”

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Ideals in atomic posets

Erné

Joshi

2015

Discrete Mathematics

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a b s t r a c tThe ''bottom'' of a partially ordered set (poset) Q is the set Q ℓ of its lower bounds (hence, Q ℓ is empty or a singleton). The poset Q is said to be atomic if each element of Q \Q ℓ dominates an atom, that is, a minimal element of Q \Q ℓ . Thus, all finite posets are atomic. We study general closure systems of down-sets (referred to as ideals) in posets. In particular, we investigate so-called m-ideals for arbitrary cardinals m, providing common generalizations of ideals in lattices and of cuts in posets. Various properties of posets and their atoms are described by means of ideals, polars (annihilators) and residuals, defined parallel to ring theory. We deduce diverse characterizations of atomic posets satisfying certain distributive laws, e.g. by the representation of specific ideals as intersections of prime ideals, or by maximality and minimality properties. We investigate non-dense ideals (down-sets having nontrivial polars) and semiprime ideals (down-sets all of whose residuals are ideals). Our results are constructive in that they do not require any set-theoretical choice principles.

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Section: Ideal Completions and Distributive Lawsmentioning

confidence: 99%

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confidence: 99%

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Ideals in atomic posets

Erné

Joshi

2015

Discrete Mathematics

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“…Rav [29] generalized Stone's Theorem to general lattices by means of semiprime ideals. More general Prime Ideal Theorems than Rav's Prime Ideal Theorem (PIT) for lattices can be found in Erné [8]. Balasubramani [1] provided many characterizations of 0-distributive lattices in terms of prime ideals.…”

Section: Introductionmentioning

confidence: 99%

Prime ideals in 0-distributive posets

Joshi

Mundlik²

2013

Open Mathematics

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In the first section of this paper, we prove an analogue of Stone's Theorem for posets satisfying DCC by using semiprime ideals. We also prove the existence of prime ideals in atomic posets in which atoms are dually distributive. Further, it is proved that every maximal non-dense (non-principal) ideal of a 0-distributive poset (meet-semilattice) is prime. The second section focuses on the characterizations of (minimal) prime ideals in pseudocomplemented posets. The third section deals with the generalization of the classical theorem of Nachbin. In fact, we prove that a dually atomic pseudocomplemented, 1-distributive poset is complemented if and only if the poset of prime ideals is unordered. In the last section, we have characterized 0-distributive posets by means of prime ideals and minimal prime ideals. MSC:06B10, 06A12

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“…The notion of a ρ-distributor was motivated also by Rav's notion of a semiprime filter [18] and Erné's notion of a distributor [6,7]. . .…”

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More on the Strength of Engeler’s Lemma

Paseka

2008

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We introduce semilattices equipped with a partial n-ary operation ρ. A useful separation lemma for such nontrivial complete partial ρ-semilattices is proved equivalent to PIT, the prime ideal theorem. The relation of various versions of the Lemma to each other and to PIT is also explored.Keywords Prime ideal theorem · Partial ρ-semilattice · ρ-prime element · Engeler's lemma Mathematics Subject Classifications (2000)Motivated by Banaschewski's prime element theorem [1], the results from [2] and our recent result [16] that PIT, the prime ideal theorem implies that any element outside some Scott-open ρ-distributor S in a nontrivial complete partial m-semilattice is below an ρ-prime element outside S, this note presents an extension of our result from partial binary operations to partial n-ary operations.Therefore we introduce semilattices equipped with a partial n-ary operation ρ. A useful separation lemma for such nontrivial complete partial ρ-semilattices is proved equivalent to PIT. The relation of various versions of the Lemma to each other and to PIT is also explored.In addition, we prove that PIT implies ρ-spatiality for any ρ-semidistributive algebraic partial cρ-lattice. Finally, we look at the role of conjunctivity with respect to ρ-spatiality of a partial cρ-lattice.

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Prime Ideal Theory for General Algebras

Cited by 7 publications

References 18 publications

Ideals in atomic posets

Ideals in atomic posets

Prime ideals in 0-distributive posets

More on the Strength of Engeler’s Lemma

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