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Semigroups of real numbers under the min/max operators #1631

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min/max semigroups of real numbers
malarbol Oct 25, 2025
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Update src/real-numbers/binary-maximum-real-numbers.lagda.md
malarbol Oct 30, 2025
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Update src/real-numbers/binary-minimum-real-numbers.lagda.md
malarbol Oct 30, 2025
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Merge branch 'master' into min/max-large-semigroups-R
malarbol Oct 30, 2025
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75 changes: 75 additions & 0 deletions src/real-numbers/binary-maximum-real-numbers.lagda.md
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Original file line number Diff line number Diff line change
Expand Up @@ -22,6 +22,9 @@ open import foundation.propositional-truncations
open import foundation.propositions
open import foundation.universe-levels

open import group-theory.large-semigroups
open import group-theory.semigroups

open import metric-spaces.metric-space-of-short-functions-metric-spaces
open import metric-spaces.short-functions-metric-spaces

Expand Down Expand Up @@ -200,6 +203,78 @@ module _
( is-least-binary-upper-bound-max-ℝ y x)))
```

### The binary maximum is associative

```agda
module _
{l1 l2 l3 : Level}
(x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
where

abstract
associative-max-ℝ : max-ℝ (max-ℝ x y) z ＝ max-ℝ x (max-ℝ y z)
associative-max-ℝ =
Comment on lines +215 to +216
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@fredrik-bakke fredrik-bakke Oct 28, 2025 •
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This is true in any large poset with meets, by the same argument as you demonstrate. Would you be willing to prove it in that generality, if it is not already?

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@fredrik-bakke fredrik-bakke Oct 28, 2025

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As a corollary, any large poset has an associated large semigroup under max.

antisymmetric-leq-ℝ
( max-ℝ (max-ℝ x y) z)
( max-ℝ x (max-ℝ y z))
( leq-max-leq-leq-ℝ
( max-ℝ x y)
( z)
( max-ℝ x (max-ℝ y z))
( leq-max-leq-leq-ℝ
( x)
( y)
( max-ℝ x (max-ℝ y z))
( leq-left-max-ℝ x (max-ℝ y z))
( transitive-leq-ℝ
( y)
( max-ℝ y z)
( max-ℝ x (max-ℝ y z))
( leq-right-max-ℝ x (max-ℝ y z))
( leq-left-max-ℝ y z)))
( transitive-leq-ℝ
( z)
( max-ℝ y z)
( max-ℝ x (max-ℝ y z))
( leq-right-max-ℝ x (max-ℝ y z))
( leq-right-max-ℝ y z)))
( leq-max-leq-leq-ℝ
( x)
( max-ℝ y z)
( max-ℝ (max-ℝ x y) z)
( transitive-leq-ℝ
( x)
( max-ℝ x y)
( max-ℝ (max-ℝ x y) z)
( leq-left-max-ℝ (max-ℝ x y) z)
( leq-left-max-ℝ x y))
( leq-max-leq-leq-ℝ
( y)
( z)
( max-ℝ (max-ℝ x y) z)
( transitive-leq-ℝ
( y)
( max-ℝ x y)
( max-ℝ (max-ℝ x y) z)
( leq-left-max-ℝ (max-ℝ x y) z)
( leq-right-max-ℝ x y))
( leq-right-max-ℝ (max-ℝ x y) z)))
```

### The large semigroup of real numbers under the maximum operator

```agda
large-semigroup-max-ℝ : Large-Semigroup lsuc
large-semigroup-max-ℝ = make-Large-Semigroup ℝ-Set max-ℝ associative-max-ℝ
```

### The semigroup of real numbers under the maximum operator at a given level

```agda
semigroup-max-ℝ : (l : Level) → Semigroup (lsuc l)
semigroup-max-ℝ = semigroup-Large-Semigroup large-semigroup-max-ℝ
```

### The large poset of real numbers has joins

```agda
Expand Down
75 changes: 75 additions & 0 deletions src/real-numbers/binary-minimum-real-numbers.lagda.md
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Original file line number Diff line number Diff line change
Expand Up @@ -19,6 +19,9 @@ open import foundation.logical-equivalences
open import foundation.transport-along-identifications
open import foundation.universe-levels

open import group-theory.large-semigroups
open import group-theory.semigroups

open import order-theory.greatest-lower-bounds-large-posets
open import order-theory.large-meet-semilattices
open import order-theory.meet-semilattices
Expand Down Expand Up @@ -132,6 +135,78 @@ module _
( is-greatest-binary-lower-bound-min-ℝ x y))
```

### The binary minimum is associative

```agda
module _
{l1 l2 l3 : Level}
(x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
where

abstract
associative-min-ℝ : min-ℝ (min-ℝ x y) z ＝ min-ℝ x (min-ℝ y z)
associative-min-ℝ =
antisymmetric-leq-ℝ
( min-ℝ (min-ℝ x y) z)
( min-ℝ x (min-ℝ y z))
Comment on lines +147 to +151
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@fredrik-bakke fredrik-bakke Oct 28, 2025

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Same comment here. this is true by the same proof that you write in all large posets with joins.

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@fredrik-bakke fredrik-bakke Oct 28, 2025

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As a corollary, any large poset has an associated large semigroup under min.

( leq-min-leq-leq-ℝ
( x)
( min-ℝ y z)
( min-ℝ (min-ℝ x y) z)
( transitive-leq-ℝ
( min-ℝ (min-ℝ x y) z)
( min-ℝ x y)
( x)
( leq-left-min-ℝ x y)
( leq-left-min-ℝ (min-ℝ x y) z))
( leq-min-leq-leq-ℝ
( y)
( z)
( min-ℝ ( min-ℝ x y) z)
( transitive-leq-ℝ
( min-ℝ (min-ℝ x y) z)
( min-ℝ x y)
( y)
( leq-right-min-ℝ x y)
( leq-left-min-ℝ (min-ℝ x y) z))
( leq-right-min-ℝ (min-ℝ x y) z)))
( leq-min-leq-leq-ℝ
( min-ℝ x y)
( z)
( min-ℝ x (min-ℝ y z))
( leq-min-leq-leq-ℝ
( x)
( y)
( min-ℝ x (min-ℝ y z))
( leq-left-min-ℝ x (min-ℝ y z))
( transitive-leq-ℝ
( min-ℝ x (min-ℝ y z))
( min-ℝ y z)
( y)
( leq-left-min-ℝ y z)
( leq-right-min-ℝ x (min-ℝ y z))))
( transitive-leq-ℝ
( min-ℝ x (min-ℝ y z))
( min-ℝ y z)
( z)
( leq-right-min-ℝ y z)
( leq-right-min-ℝ x (min-ℝ y z))))
```

### The large semigroup of real numbers under the minimum operator

```agda
large-semigroup-min-ℝ : Large-Semigroup lsuc
large-semigroup-min-ℝ = make-Large-Semigroup ℝ-Set min-ℝ associative-min-ℝ
```

### The semigroup of real numbers under the minimum operator at a given level

```agda
semigroup-min-ℝ : (l : Level) → Semigroup (lsuc l)
semigroup-min-ℝ = semigroup-Large-Semigroup large-semigroup-min-ℝ
```

### The large poset of real numbers has meets

```agda
Expand Down