diff options
Diffstat (limited to 'src/algebra/catdef.spad.pamphlet')
| -rw-r--r-- | src/algebra/catdef.spad.pamphlet | 36 |
1 files changed, 15 insertions, 21 deletions
diff --git a/src/algebra/catdef.spad.pamphlet b/src/algebra/catdef.spad.pamphlet index 32f1f3c7..735d2e43 100644 --- a/src/algebra/catdef.spad.pamphlet +++ b/src/algebra/catdef.spad.pamphlet @@ -20,18 +20,16 @@ ++ Date Created: May 31, 2009 ++ Date Last Modified: May 31, 2009 ++ Description: -++ A set is an \spad{R}-left linear set if it is stable by left-dilation -++ by elements in the ring \spad{R}. This category differs from -++ \spad{LeftModule} in that no other assumption (such as addition) -++ is made about the underlying set. +++ A set is an \spad{S}-left linear set if it is stable by left-dilation +++ by elements in the semigroup \spad{S}. ++ See Also: RightLinearSet. -LeftLinearSet(R: Rng): Category == SetCategory with +LeftLinearSet(S: SemiGroup): Category == SetCategory with 0: % ++ \spad{0} represents the origin of the linear set zero?: % -> Boolean - ++ \spad{zero? x} holds is \spad{x} is the origin. - *: (R,%) -> % - ++ \spad{r*x} is the left-dilation of \spad{x} by \spad{r}. + ++ \spad{zero? x} holds if \spad{x} is the origin. + *: (S,%) -> % + ++ \spad{s*x} is the left-dilation of \spad{x} by \spad{s}. @ @@ -41,18 +39,16 @@ LeftLinearSet(R: Rng): Category == SetCategory with ++ Date Created: May 31, 2009 ++ Date Last Modified: May 31, 2009 ++ Description: -++ A set is an \spad{R}-right linear set if it is stable by right-dilation -++ by elements in the ring \spad{R}. This category differs from -++ \spad{RightModule} in that no other assumption (such as addition) -++ is made about the underlying set. +++ A set is an \spad{S}-right linear set if it is stable by right-dilation +++ by elements in the semigroup \spad{S}. ++ See Also: LeftLinearSet. -RightLinearSet(R: Rng): Category == SetCategory with +RightLinearSet(S: SemiGroup): Category == SetCategory with 0: % ++ \spad{0} represents the origin of the linear set zero?: % -> Boolean - ++ \spad{zero? x} holds is \spad{x} is the origin. - *: (%,R) -> % - ++ \spad{r*x} is the left-dilation of \spad{x} by \spad{r}. + ++ \spad{zero? x} holds if \spad{x} is the origin. + *: (%,S) -> % + ++ \spad{x*s} is the right-dilation of \spad{x} by \spad{s}. @ @@ -62,12 +58,10 @@ RightLinearSet(R: Rng): Category == SetCategory with ++ Date Created: May 31, 2009 ++ Date Last Modified: May 31, 2009 ++ Description: -++ A set is an \spad{R}-linear set if it is stable by dilation -++ by elements in the ring \spad{R}. This category differs from -++ \spad{Module} in that no other assumption (such as addition) -++ is made about the underlying set. +++ A set is an \spad{S}-linear set if it is stable by dilation +++ by elements in the semigroup \spad{S}. ++ See Also: LeftLinearSet, RightLinearSet. -LinearSet(R: Rng): Category == Join(LeftLinearSet R, RightLinearSet R) +LinearSet(S: SemiGroup): Category == Join(LeftLinearSet S, RightLinearSet S) @ |