# Proposals for additions to Unicode

## N-ARY RESTRICTED PRODUCT

The N-ARY RESTRICTED PRODUCT character should be a hybrid between U+220F N-ARY PRODUCT (∏, a kind of capital Π) and U+2210 N-ARY COPRODUCT (∐, the same glyph, inverted), having the top half of the former and the bottom half of the latter (so that it is symmetric around its median horizontal axis).

and combine to make

It is typically used in mathematics to designate some kind of “restricted” product of an infinite family, meaning the elements of the product which belong to some implicitly understood subset of the factors for all but finitely many factor. Examples of uses are the construction of the adèle ring or idèle group of an algebraic number field. Other notations for the restricted product include ∏′, but the symbol whose addition is proposed here, who was introduced by John Tate, seems most elegant and should be added to Unicode.

Examples of uses of the symbol in printed books:

• Jean-Pierre Serre, Galois Cohomology, Springer (1997), ISBN 3-540-61990-9, sections 6.1 through 6.3, at various places between pages 106 and 108, in describing the Poitou-Tate exact sequence; on page 106, Serre claims that the symbol was suggested by Tate to emphasize that it is intermediate between a product and a sum. [Excerpt scan here.]
• Jürgen Neukirch, Algebraic Number Theory, Springer Grundlehren 322 (1999), ISBN 3-540-65399-6, at the beginning of chapter VI, p. 357, when defining the adele group of a number field. [Exceprt scan here.]

The character is (apparently) missing from the LaTeX “comprehensive symbol list”, however, so I assume no LaTeX package has it.

## N-ARY (large) variants of squared operators

Squared operators, at least U+229E SQUARED PLUS and U+22A0 SQUARED TIMES, and probably also U+22A1 SQUARED DOT OPERATOR, should have an N-ARY OPERATOR (i.e., large operator) variant, in the same way that U+2A01 N-ARY CIRCLED PLUS OPERATOR (⨁) and U+2A02 N-ARY CIRCLED TIMES OPERATOR (⨂) and U+2A00 N-ARY CIRCLED DOT OPERATOR (⨀) are large operator variants of U+2295 CIRCLED PLUS (⊕) and U+2297 CIRCLED TIMES (⊗) and U+2299 CIRCLED DOT OPERATOR (⊙).

The large operator variant differs from the small symbol not merely in size but also in semantics: whereas the small operator is binary, meaning that it connects two symbols or expressions, the large operator is N-ary, that is, it is followed by an indexed expression, and typically itself bears an index and exponent to indicate the range of the index set. In other words, the large operator variant is to the small one as U+2211 N-ARY SUMMATION (∑) is to simple binary addition (U+002B PLUS SIGN, +).

The use of the proposed N-ARY SQUARED TIMES OPERATOR at least should be easy to find in the mathematical literature in algebraic geometry to denote the external tensor product of a family. See for example SGA4½, exp. VI(2.4).

These operators are available as `\bigboxplus`, `\bigboxtimes` and `\bigboxdot` in the LaTeX `mathabx` package.

## CIRCLED LOGICAL AND and CIRCLED LOGICAL OR

For some reason, Unicode has U+27CE SQUARED LOGICAL AND and U+27CF SQUARED LOGICAL OR but not the obvious CIRCLED LOGICAL AND and CIRCLED LOGICAL OR (and perhaps corresponding “large” versions) that are are available as `\circledwedge`, `\circledvee` in the LaTeX `txfonts` and `pxfonts` packages.

(This is highly surprising since Unicode has a considerable number of variations around the wedge and vee signs, like U+2A51 LOGICAL AND WITH DOT ABOVE or U+2A53 DOUBLE LOGICAL AND or U+2A55 TWO INTERSECTING LOGICAL AND or U+2A5A LOGICAL AND WITH MIDDLE STEM or U+2A5E LOGICAL AND WITH DOUBLE OVERBAR and corresponding “or” signs, but not the “circled” form. We need to find some texts that use these.)

## MATHEMATICAL DOUBLE-STRUCK SMALL MU

There is, of course, reason to think that one should add the entire blackboard-bold Greek alphabet, but at the very least the proposed MATHEMATICAL DOUBLE-STRUCK SMALL MU has found its way into mathematical texts as a notation for the group of roots of unity.

See, for example, Milne, Étale cohomology, ISBN 0-691-08238-3, page (xiii) in the introduction and throughout the book. [The page in the introduction is viewable from the Look inside feature on the Amazon page for the book.] The use of blackboard bold in the same line and elsewhere in the page makes it clear that the typographically poor symbol is supposed to be a blackboard bold μ. The same place makes use of a MATHEMATICAL DOUBLE-STRUCK SMALL ALPHA, which is arguably less common.

The character is apparently missing from LaTeX packages.

## Weather symbols

Internationally standardized symbols for various meteorological conditions (e.g., cloud genus, precipitation, etc.) are used on on weather charts and weather reports. They are defined in appendix II-4 of the World Meteorological Organization's Manual on the Global Data-processing and Forecasting System (volume I) and form a closed set of (ca. 150) symbols with well-defined semantics that I believe are appropriate for inclusion in Unicode (and some of them are already present, e.g., U+2608 THUNDERSTORM). More debatable would be which of these symbols need to be unified with existing Unicode characters.

According to this presentation, there may already have been a discussion about whether to include these characters in Unicode: this thread back in 2003 suggests that it probably did not go very far, possibly for lack of appropriate references. The question will inevitably arise as to whether the symbols in question are indeed characters, or merely glyphs.