What’s this plant please?

[Eric Hardy]

Please can someone identify the garden plant for me. We have had it for years but it’s name escapes me. It is growing out of a joint in some steps and spreads like a mat over the steps.
Here it is in situ



And these two I tried to photograph like a specimen.

[Palustris]

Phuopsis stylosa at a guess.

[Eric Hardy]

Thanks Eric.

[ideasguy]

You are a genius Eric. I didn't know that plant beforehand. It was actually in the Ideas Genie Pro 6 Master Database as a species and one cultivar -  Phuopsis stylosa 'Purpurea'.
Using that to google search, I found this website which gives a good description:
https://www.rushfields.com/plantguide/plant/perennial/phuopsis-stylosa-purpurea