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operation ⋅ that sends any two elements a, b ∈ M to another element, a⋅b∈M. The symbol ⋅ is a general placeholder for a properly defined operation. This requirement that for all a, b in M, the result of the operation a⋅b also be in M, is known as the magma or closure property. +::: + + +::: {.Definition #semigroup title="semigroup"} + +To understand semigroup, you need to know magma... + +:::: {#incmagma include="magma"} +::::: + +... Now you know magma, let's see the definition of semigroup. + +(S, ⋅) is a semigroup if it is an associative magma. +::: + + +::: {.Definition #monoid title="monoid"} +To understand monoid, you need to know semigroup... + +:::: {#incsemigroup include="semigroup"} +::::: + +... Now you know semigroup, let's see the definition of monoid. + +A monoid is a semigroup with an identity element. +::: + + +::: {.Definition #group title="group"} +To understand group, you need to know monoid... + +:::: {#incmonoid include="monoid"} +::::: + +... Now you know monoid, let's see the definition of group. + +(S, ⋅) is a group is it is a monoid such that every element has an unique inverse. +::: diff --git a/algebraic_structures.processed.md b/algebraic_structures.processed.md new file mode 100644 index 0000000..d59d20e --- /dev/null +++ b/algebraic_structures.processed.md @@ -0,0 +1,92 @@ +::: {#magma .Definition title="magma"} +A magma is a set M with an operation ⋅ that sends any two elements a, b +∈ M to another element, a⋅b∈M. The symbol ⋅ is a general placeholder for +a properly defined operation. This requirement that for all a, b in M, +the result of the operation a⋅b also be in M, is known as the magma or +closure property. +::: + +::::: {#semigroup .Definition title="semigroup"} +To understand semigroup, you need to know magma... + +:::: {#incmagma include="magma"} +::: {.Definition title="magma"} +A magma is a set M with an operation ⋅ that sends any two elements a, b +∈ M to another element, a⋅b∈M. The symbol ⋅ is a general placeholder for +a properly defined operation. This requirement that for all a, b in M, +the result of the operation a⋅b also be in M, is known as the magma or +closure property. +::: +:::: + +... Now you know magma, let's see the definition of semigroup. + +(S, ⋅) is a semigroup if it is an associative magma. +::::: + +::::::: {#monoid .Definition title="monoid"} +To understand monoid, you need to know semigroup... + +:::::: {#incsemigroup include="semigroup"} +::::: {.Definition title="semigroup"} +To understand semigroup, you need to know magma... + +:::: {#incmagma include="magma"} +::: {.Definition title="magma"} +A magma is a set M with an operation ⋅ that sends any two elements a, b +∈ M to another element, a⋅b∈M. The symbol ⋅ is a general placeholder for +a properly defined operation. This requirement that for all a, b in M, +the result of the operation a⋅b also be in M, is known as the magma or +closure property. +::: +:::: + +... Now you know magma, let's see the definition of semigroup. + +(S, ⋅) is a semigroup if it is an associative magma. +::::: +:::::: + +... Now you know semigroup, let's see the definition of monoid. + +A monoid is a semigroup with an identity element. +::::::: + +::::::::: {#group .Definition title="group"} +To understand group, you need to know monoid... + +:::::::: {#incmonoid include="monoid"} +::::::: {.Definition title="monoid"} +To understand monoid, you need to know semigroup... + +:::::: {#incsemigroup include="semigroup"} +::::: {.Definition title="semigroup"} +To understand semigroup, you need to know magma... + +:::: {#incmagma include="magma"} +::: {.Definition title="magma"} +A magma is a set M with an operation ⋅ that sends any two elements a, b +∈ M to another element, a⋅b∈M. The symbol ⋅ is a general placeholder for +a properly defined operation. This requirement that for all a, b in M, +the result of the operation a⋅b also be in M, is known as the magma or +closure property. +::: +:::: + +... Now you know magma, let's see the definition of semigroup. + +(S, ⋅) is a semigroup if it is an associative magma. +::::: +:::::: + +... Now you know semigroup, let's see the definition of monoid. + +A monoid is a semigroup with an identity element. +::::::: +:::::::: + +... Now you know monoid, let's see the definition of group. + +(S, ⋅) is a group is it is a monoid such that every element has an +unique inverse. +:::::::::