### Abstract

The space of maximal d-ideals of C(X) is well-known and is widely studied. It is known that the space of maximal d-ideals is homeomorphic to the Z^{♯}(X)-ultrafilters, and this space is the minimal quasi F-cover of a compact Tychonoff space X. In the current article we generalize this concept for M-frames, algebraic frames with the finite intersection property. In particular, we explore various properties of the maximal d-elements of a frame L, and their relation with the ultrafilters of 픎L^{⊥}, the polars of the compact elements of L. On a separate note, we revisit the Lemma on Ultrafilters and establish the correspondence between the minimal prime elements spaces of L with the spaces of ultrafilters of 픎L. Finally, we show that for complemented frames L, Min(L) = Max(dL), a result parallel to the one known for Riesz spaces, W-objects, and topological spaces.

Original language | English (US) |
---|---|

Pages (from-to) | 377-390 |

Number of pages | 14 |

Journal | Order |

Volume | 36 |

Issue number | 2 |

DOIs | |

State | Published - Jul 15 2019 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Geometry and Topology
- Computational Theory and Mathematics

### Cite this

*Order*,

*36*(2), 377-390. https://doi.org/10.1007/s11083-018-9472-5

}

*Order*, vol. 36, no. 2, pp. 377-390. https://doi.org/10.1007/s11083-018-9472-5

**Maximal d-Elements of an Algebraic Frame.** / Bhattacharjee, Papiya.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Maximal d-Elements of an Algebraic Frame

AU - Bhattacharjee, Papiya

PY - 2019/7/15

Y1 - 2019/7/15

N2 - The space of maximal d-ideals of C(X) is well-known and is widely studied. It is known that the space of maximal d-ideals is homeomorphic to the Z♯(X)-ultrafilters, and this space is the minimal quasi F-cover of a compact Tychonoff space X. In the current article we generalize this concept for M-frames, algebraic frames with the finite intersection property. In particular, we explore various properties of the maximal d-elements of a frame L, and their relation with the ultrafilters of 픎L⊥, the polars of the compact elements of L. On a separate note, we revisit the Lemma on Ultrafilters and establish the correspondence between the minimal prime elements spaces of L with the spaces of ultrafilters of 픎L. Finally, we show that for complemented frames L, Min(L) = Max(dL), a result parallel to the one known for Riesz spaces, W-objects, and topological spaces.

AB - The space of maximal d-ideals of C(X) is well-known and is widely studied. It is known that the space of maximal d-ideals is homeomorphic to the Z♯(X)-ultrafilters, and this space is the minimal quasi F-cover of a compact Tychonoff space X. In the current article we generalize this concept for M-frames, algebraic frames with the finite intersection property. In particular, we explore various properties of the maximal d-elements of a frame L, and their relation with the ultrafilters of 픎L⊥, the polars of the compact elements of L. On a separate note, we revisit the Lemma on Ultrafilters and establish the correspondence between the minimal prime elements spaces of L with the spaces of ultrafilters of 픎L. Finally, we show that for complemented frames L, Min(L) = Max(dL), a result parallel to the one known for Riesz spaces, W-objects, and topological spaces.

UR - http://www.scopus.com/inward/record.url?scp=85053622177&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85053622177&partnerID=8YFLogxK

U2 - 10.1007/s11083-018-9472-5

DO - 10.1007/s11083-018-9472-5

M3 - Article

AN - SCOPUS:85053622177

VL - 36

SP - 377

EP - 390

JO - Order

JF - Order

SN - 0167-8094

IS - 2

ER -