On Complete Representations of Reducts of Polyadic Algebras

Tarek Sayed Ahmed

doi:10.1007/s11225-008-9131-8

On complete representations and minimal completions in algebraic logic, both positive and negative results

Bulletin of the Section of Logic ◽

10.18778/0138-0680.2021.17 ◽

2021 ◽

Author(s):

Tarek Sayed Ahmed

Keyword(s):

Algebraic Logic ◽

Relation Algebras ◽

Cylindric Algebras ◽

Polyadic Algebras ◽

Negative Results ◽

Complete Representations

Fix a finite ordinal $n\geq 3$ and let $\alpha$ be an arbitrary ordinal. Let $\mathsf{CA}_n$ denote the class of cylindric algebras of dimension $n$ and $\sf RA$ denote the class of relation algebras. Let $\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})$ stand for the class of polyadic (equality) algebras of dimension $\alpha$. We reprove that the class $\mathsf{CRCA}_n$ of completely representable $\mathsf{CA}_n$s, and the class \(\sf CRRA$ of completely representable $\mathsf{RA}$s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety $\sf V$ between polyadic algebras of dimension $n$ and diagonal free $\mathsf{CA}_n$s. We show that that the class of completely and strongly representable algebras in $\sf V$ is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class $\sf CRRA$ is not closed under $\equiv_{\infty,\omega}$. In contrast, we show that given $\alpha\geq \omega$, and an atomic $\mathfrak{A}\in \mathsf{PEA}_{\alpha}$, then for any $n<\omega$, $\mathfrak{Nr}_n\A$ is a completely representable $\mathsf{PEA}_n$. We show that for any $\alpha\geq \omega$, the class of completely representable algebras in certain reducts of $\mathsf{PA}_{\alpha}$s, that happen to be varieties, is elementary. We show that for $\alpha\geq \omega$, the the class of polyadic-cylindric algebras dimension $\alpha$, introduced by Ferenczi, the completely representable algebras (slightly altering representing algebras) coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension $n$ and $\mathsf{PA}_{\alpha}$s for $\alpha$ an infinite ordinal, proving negative results for the first and positive ones for the second.

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On automorphisms of polyadic algebras

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1964-0162741-6 ◽

1964 ◽

Vol 112 (1) ◽

pp. 84-84 ◽

Cited By ~ 9

Author(s):

Aubert Daigneault

Keyword(s):

Polyadic Algebras

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An autobiography of polyadic algebras

Logic Journal of IGPL ◽

10.1093/jigpal/8.4.383 ◽

2000 ◽

Vol 8 (4) ◽

pp. 383-392 ◽

Cited By ~ 4

Author(s):

P. Halmos

Keyword(s):

Polyadic Algebras

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EXOTIC BIRCH-LEAFMINING SAWFLIES (HYMENOPTERA: TENTHREDINIDAE) IN ALBERTA: DISTRIBUTIONS, SEASONAL ACTIVITIES, AND THE POTENTIAL FOR COMPETITION

The Canadian Entomologist ◽

10.4039/ent129319-2 ◽

1997 ◽

Vol 129 (2) ◽

pp. 319-333 ◽

Cited By ~ 11

Author(s):

Scott C. Digweed ◽

John R. Spence ◽

David W. Langor

Keyword(s):

Urban Trees ◽

Seasonal Activity ◽

Mature Trees ◽

Sticky Traps ◽

Rural Location ◽

Three Generations ◽

Complete Representations ◽

Fenusa Pusilla ◽

Profenusa Thomsoni ◽

Southern Alberta

AbstractThe exotic birch-leafmining sawflies Fenusa pusilla (Lepeletier), Profenusa thomsoni (Konow), and Heterarthrus nemoratus (Fallen) occurred in Alberta during 1992–1995, but only the first two were abundant. Birch-leafmining sawflies occurred at all sites surveyed in central and southern Alberta, and appeared to be expanding their ranges northward. Adult F. pusilla began emerging in mid-May (approximately 220 DD05), and there were one to three generations per year, depending on location and year. Female F. pusilla were relatively less abundant on young than on mature trees. Profenusa thomsoni began attacking trees in June (between 400 and 550 DD05), and was invariably univoltine. Both species were more abundant and were active earlier on urban trees than at a nearby rural location. The highest catches and most complete representations of seasonal activity were obtained using yellow sticky traps. Larval F. pusilla and P. thomsoni are unlikely to compete directly for leaf resources because their leafmining activities are separated spatially and temporally, but they probably compete intraspecifically.

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